the possible contributions from the different hyperfine levels of the relevant excited states and depend on the detuning as well. The quantum mechanical Stokes vector components describing the polarization state of light are $$\begin{gathered}
 \nonumber
 \hat s_+ = \hat s_x + i \hat s_y = \hat a_R^\dag \hat a_L,\\
 \nonumber
 \hat s_z = (\hat  a_R^\dag \hat a_R - \hat  a_L^\dag \hat a_L)/2,\\
 \hat \phi = \hat  a_R^\dag \hat a_R + \hat  a_L^\dag \hat a_L,
 \label{eq:Stokes}\end{gathered}$$ where $\hat{a}_R$ and $\hat{a}_L$ are annihilation operators for right and left circular polarized, spatially localized photons, and they obey the standard commutation relation $[\hat a_i(z), \hat a_{j}(z')] = c \delta_{ij} \delta (z-z')$.

The photon flux $\hat \phi$ commutes with any other operator in Eq.  and is usually treated as a c-number. The quadratic $\hat F_x^{(j)2}$ component can be compensated with the second order Zeeman shift, $$\begin{gathered}
 \label{eq:2nd-Zeeman}
 \hat H_Z^{(j)} = \omega_L \hat F_x^{(j)} + \beta \hat F_x^{(j)2},\end{gathered}$$ by tuning the magnetic field such that $\beta = g_2 \phi$ [@PhysRevLett.99.163002; @PhysRevLett.101.073601]. To meet the two-photon resonance condition, the right and left polarized light components have to oscillate on the two Larmor sidebands. After summing over all atoms and performing the rotating wave transformation, we have $$\begin{gathered}
 