  The Rabi model that describes the fundamental interaction between a two-level
system with a quantized harmonic oscillator is one of the simplest and most
ubiquitous models in modern physics. However, this model has not been solved
exactly because it is hard to find a second conserved quantity besides the
energy. Here we present a unitary transformation to map this unsolvable Rabi
model into a solvable Jaynes-Cummings-like model by choosing a proper variation
parameter. As a result, the analytical energy spectrums and wavefunctions
including both the ground and the excited states can be obtained easily.
Moreover, these explicit results agree well with the direct numerical
simulations in a wide range of the experimental parameters. In addition, based
on our obtained energy spectrums, the recent experimental observation of
Bloch-Siegert in the circuit quantum electrodynamics with the ultrastrong
coupling can be explained perfectly. Our results have the potential application
in the solid-state quantum information processing.
