  In circuit quantum electrodynamics (circuit QED), an artificial "circuit
atom" can couple to a quantized microwave radiation much stronger than its real
atomic counterpart. The celebrated quantum Rabi model describes the simplest
interaction of a two-level system with a single-mode boson field. When the
coupling is large enough, the bare multilevel structure of a realistic circuit
atom cannot be ignored even if the circuit is strongly anharmonic. We explored
this situation theoretically for flux (fluxonium) and charge (Cooper pair box)
type multi-level circuits tuned to their respective flux/charge degeneracy
points. We identified which spectral features of the quantum Rabi model survive
and which are renormalized for large coupling. Despite significant
renormalization of the low-energy spectrum in the fluxonium case, the key
quantum Rabi feature -- nearly-degenerate vacuum consisting of an atomic state
entangled with a multi-photon field -- appears in both types of circuits when
the coupling is sufficiently large. Like in the quantum Rabi model, for very
large couplings the entanglement spectrum is dominated by only two, nearly
equal eigenvalues, in spite of the fact that a large number of bare atomic
states are actually involved in the atom-resonator ground state. We interpret
the emergence of the two-fold degeneracy of the vacuum of both circuits as an
environmental suppression of flux/charge tunneling due to their dressing by
virtual low-/high-impedance photons in the resonator. For flux tunneling, the
dressing is nothing else than the shunting of a Josephson atom with a large
capacitance of the resonator. Suppression of charge tunneling is a
manifestation of the dynamical Coulomb blockade of transport in tunnel
junctions connected to resistive leads.
