Bloch-Siegert shift of the Rabi model

We apply a simple analytical method based on a unitary transformation to calculate the Bloch-Siegert (BS) shift over the entire driving-strength range. In quantitative comparison with the numerically exact BS shift obtained by Floquet formalism as well as the previous BS results, we confirm that our calculated results are not only accurate in the weak-driving regime but also correct in the strong-driving limit. In the intermediate strong-driving regime, the calculated values of the BS shift are nearly the same as the exact ones. It turns out that our calculation for the BS shift is beyond perturbation. Meanwhile, we demonstrate the signatures caused by the BS shift by monitoring the excited-state population and the probe-pump spectrum under the experiment accessible conditions. In particular, we find that when the driving frequency is fixed at the transition frequency of the system, the line shape of the probe-pump spectrum becomes asymmetric with the increase of the driving strength, which may be verified experimentally.

Figure 1

The BS shift as a function of driving strength $A$ for various methods. The precision of numerical method is defined by the deviation of $\overline{P}$ from the maximum $1/2$, which is about ${10}^{-15}$ for given $A$ and ${\omega }_{\mathrm{res}}$ in this work.

Figure 2

The deviation between analytical results and numerical results.

Figure 3

The time-averaged population $\overline{{\rho }_{++}}$ as a function of driving frequency $\omega$ for $\kappa =2\times {10}^{-3}{\omega }_0$ and various driving strength $A$.

Figure 4

The probe-pump spectrum $S\left(\upsilon \right)$ as a function of probe frequency $\upsilon$ for $\kappa =2\times {10}^{-3}{\omega }_0$.

Figure 5

The probe-pump spectrum $S\left(\upsilon \right)$ as a function of probe frequency $\upsilon$ for $\kappa =2\times {10}^{-3}{\omega }_0$ with $A=0.1{\omega }_0$.