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We investigate the bias-modulated dynamics of a strongly driven two-levelsystem using the counter-rotating-hybridized rotating-wave (CHRW) method. ThisCHRW method treats the driving field and the bias on equal footing by a unitarytransformation with two parameters $\xi$ and $\zeta$, and is nonperturbative indriving strength, tunneling amplitude or bias. In addition, this CHRW method isbeyond the traditional rotating-wave approximation (Rabi-RWA) and yet byproperly choosing the two parameters $\xi$ and $\zeta$, the transformedHamiltonian takes the RWA form with a renormalized energy splitting and arenormalized driving strength. The reformulated CHRW method possesses the samemathematical simplicity as the Rabi-RWA approach and thus allows us tocalculate analytically the dynamics and explore explicitly the effect of thebias. We show that the CHRW method gives the accurate driven dynamics for awide range of parameters as compared to the numerically exact results. Whenenergy scales of the driving are comparable to the intrinsic energy scale ofthe two-level systems, the counter-rotating interactions and static biasprofoundly influence the generalized Rabi frequency. In this regime, whereordinary perturbation approaches fail, the CHRW works very well andefficiently. We also demonstrate the dynamics of the system in thestrong-driving and off-resonance cases for which the Rabi-RWA method breaksdown but the CHRW method remains valid. We obtain analytical expressions forthe generalized Rabi frequency and bias-modulated Bloch-Siegert shift asfunctions of the bias, tunneling and driving field parameters. The CHRWapproach is a mathematically simple and physically clear method. It can beapplied to treat some complicated problems for which a numerical study isdifficult to perform.

Bias-modulated dynamics of a strongly driven two-level system