Effects of counter-rotating interaction on driven tunneling dynamics: Coherent destruction of tunneling and Bloch-Siegert shift

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.. 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.