  We investigate the bias-modulated dynamics of a strongly driven two-level
system using the counter-rotating-hybridized rotating-wave (CHRW) method. This
CHRW method treats the driving field and the bias on equal footing by a unitary
transformation with two parameters $\xi$ and $\zeta$, and is nonperturbative in
driving strength, tunneling amplitude or bias. In addition, this CHRW method is
beyond the traditional rotating-wave approximation (Rabi-RWA) and yet by
properly choosing the two parameters $\xi$ and $\zeta$, the transformed
Hamiltonian takes the RWA form with a renormalized energy splitting and a
renormalized driving strength. The reformulated CHRW method possesses the same
mathematical simplicity as the Rabi-RWA approach and thus allows us to
calculate analytically the dynamics and explore explicitly the effect of the
bias. We show that the CHRW method gives the accurate driven dynamics for a
wide range of parameters as compared to the numerically exact results. When
energy scales of the driving are comparable to the intrinsic energy scale of
the two-level systems, the counter-rotating interactions and static bias
profoundly influence the generalized Rabi frequency. In this regime, where
ordinary perturbation approaches fail, the CHRW works very well and
efficiently. We also demonstrate the dynamics of the system in the
strong-driving and off-resonance cases for which the Rabi-RWA method breaks
down but the CHRW method remains valid. We obtain analytical expressions for
the generalized Rabi frequency and bias-modulated Bloch-Siegert shift as
functions of the bias, tunneling and driving field parameters. The CHRW
approach is a mathematically simple and physically clear method. It can be
applied to treat some complicated problems for which a numerical study is
difficult to perform.
