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

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 $ξ$ and $ζ$, 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 $ξ$ and $ζ$, 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.