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I have calculated the dipole transition elements of electronic states $\langle a|D_1^m|b \rangle$ following the book of Cohen-Tannoudji (Complement $E_X$) and tried then to calculate from that the Rabi frequency $$\Omega_{a\to b}=\frac{1}{2\hbar}E_m \langle a|D_1^m|b \rangle.$$ But what is the direction of the E-field vector $E_m$ for the $\pi$, $\sigma^+$, $\sigma^-$ transitions?

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I am not 100% sure about it, but I think the directions are:

$$\vec{e}_{\sigma^+}=\frac{1}{\sqrt{2}}(\vec{e}_x+i\vec{e}_y)$$ $$\vec{e}_{\pi} =\vec{e}_z$$ $$\vec{e}_{\sigma^-}=-\frac{1}{\sqrt{2}}(\vec{e}_x-i\vec{e}_y)$$

It would be nice if people could correct any mistakes especially signs and factors.

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