Consider an atom modelled as a two level system :
$$H=\frac{\hbar \omega}{2} \sigma_z $$
$|0\rangle$ and $|1\rangle$ are the ground and excited states that span the Hilbert space.
In the Rabi oscillations study, we turn on an interaction Hamiltonian using a laser, and we define the Rabi Frequency as $\Omega=\frac{\vec{d}.\vec{E_0}}{\hbar}$ where $\vec{E_0}$ is the amplitude of the electric field of the laser, and $\vec{d}$ is the quantity :
$$\vec{d}=q\langle 0 | {\widehat{r}} | 1 \rangle $$
$\widehat{r}$ being the position operator (which is a vector operator) of the two level system.
My question is :
Do the components of the vector $\langle 0 | {\widehat{r}} | 1 \rangle$ actually depend on the frequency $\omega$ ?
If so, is there a resource where I can find this law ?
I guess the answer is true, because if I turn on a laser, it will polarize my atom. Thus $\langle 0 | {\widehat{r}} | 1 \rangle$ should depend on the strength of the laser. Then it may also depend on the frequency of the atom.
But how can I find the law (if I'm not wrong).