  We investigate the dynamics of a driven two-level system (classical Rabi
model) using the counter-rotating-hybridized rotating wave method (CHRW), which
is a simple method based on a unitary transformation with a parameter $\xi$.
This approach is beyond the traditional rotating-wave approximation (Rabi-RWA)
and more importantly, remains the RWA form with a renormalized tunneling
strength and a modified driving strength. The reformulated rotating wave method
not only possesses the same mathematical simplicity as the Rabi-RWA but also
allows us to explore the effects of counter-rotating (CR) components. We focus
on the properties of off-resonance cases for which the Rabi-RWA method breaks
down. After comparing the results of different RWA schemes and those of the
numerically exact method in a wide range of parameter regime, we show that the
CHRW method gives the accurate driven dynamics which is in good agreement with
the numerical method. Moreover, the other RWA methods appear as various
limiting cases of the CHRW method. The CHRW method reveals the effects of the
CR terms clearly by means of coherent destruction of tunneling and
Bloch-Siegert shift. Our main results are as follows: (i) the dynamics of the
coherent destruction of tunneling is explicitly given and its dependence on
$\Delta$ is clarified, which is quantitatively in good agreement with the exact
results; (ii) the CR modulated Rabi frequency and the Bloch-Siegert shift are
analytically calculated, which is the same as the exact results up to fourth
order; (iii) the validity of parameter regions of different RWA methods are
given and the comparison of dynamics of these methods are shown. Since the CHRW
approach is mathematically simple as well as tractable and physically clear, it
may be extended to some complicated problems where it is difficult to do a
numerical study.
