  We examine a two-level system coupled to a quantum oscillator, typically
representing experiments in cavity and circuit quantum electrodynamics. We show
how such a system can be treated analytically in the ultrastrong coupling
limit, where the ratio $g/\Omega$ between coupling strength and oscillator
frequency approaches unity and goes beyond. In this regime the Jaynes-Cummings
model is known to fail, because counter-rotating terms have to be taken into
account. By using Van Vleck perturbation theory to higher orders in the qubit
tunneling matrix element $\Delta$ we are able to enlarge the regime of
applicability of existing analytical treatments, including in particular also
the finite bias case. We present a detailed discussion on the energy spectrum
of the system and on the dynamics of the qubit for an oscillator at low
temperature. We consider the coupling strength $g$ to all orders, and the
validity of our approach is even enhanced in the ultrastrong coupling regime.
Looking at the Fourier spectrum of the population difference, we find that many
frequencies are contributing to the dynamics. They are gathered into groups
whose spacing depends on the qubit-oscillator detuning. Furthermore, the
dynamics is not governed anymore by a vacuum Rabi splitting which scales
linearly with $g$, but by a non-trivial dressing of the tunneling matrix
element, which can be used to suppress specific frequencies through a variation
of the coupling.
