  The anisotropic Rabi model, which was proposed recently, differs from the
original Rabi model: the rotating and counter-rotating terms are governed by
two different coupling constants. This feature allows us to vary the
counter-rotating interaction independently and explore the effects of it on
some quantum properties. In this paper, we eliminate the counter-rotating terms
approximately and obtain the analytical energy spectrums and wavefunctions.
These analytical results agree well with the numerical calculations in a wide
range of the parameters including the ultrastrong coupling regime. In the weak
counter-rotating coupling limit we find out that the counter-rotating terms can
be considered as the shifts to the parameters of the Jaynes-Cummings model.
This modification shows the validness of the rotating-wave approximation on the
assumption of near-resonance and relatively weak coupling. Moreover, the
analytical expressions of several physics quantities are also derived, and the
results show the break-down of the U(1)-symmetry and the deviation from the
Jaynes-Cummings model.
