**Abstract: **
In quantum electrodynamics (QED) the interaction between an
atom and the light in nature is governed by the fine structure
constant, and thus, its coupling strength makes the weak coupling regime. On the
other hand, cavity QED supplies much stronger interaction than the
standard QED does. That is, we can obtain the strong coupling regime for the atom
and the light in cavity QED. To get such a strong interaction, we usually
handle a two-level atom and a one-mode laser which are put into a mirror cavity
(i.e., a mirror resonator). This is a physical set-up for the standard cavity
QED. In 2001 Yu. Makhilin et al. suggested that this standard
set-up should be replaced with another one. Namely, we can respectively replace the
two-level atom, the laser, and the mirror resonator by an artificial atom, a
microwave, and a circuit resonator. Here the artificial atom is a
superconducting qubit implemented by using the Josephson junction. We can obtain a
stable two-level system in this replaced cavity QED because of non-linearity of
the Josephson junction. Moreover, being free from several micro-parameters,
we can control the physical set-up with some macro-parameters because we
realize the system of cavity QED by a superconducting circuit. This replaced
cavity QED is called the circuit QED, and it has been demonstrated experimentally
in 2004. It is remarkable that the interaction between the atom and the
light is much stronger in circuit QED than it is in the standard cavity QED.
Thus, the interaction is said to be in the ultra-strong coupling regime. The circuit
QED comes of age and it is among the hottest subjects of experimental physics
now. In my talk we apply the mathematical theory of the generalized
spin-boson model to the problem on the ground state energy of a fully coupled
model in circuit QED.