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I am learning quantum optics.

In the book written by Scully includes complex rabi frequency, when dealing with $\Delta$ level configuration, the Hamiltonian including laser-atom interacting concludes complex rabi frequency.

However, from the course I took as undergraduate, where rabi frequency are treated as real numbers (vectors), I cannot see why rabi frequency has imaginary part.

Could anyone help me with this? Why it has imaginary part? And how to deduce it?

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The Rabi frequency is defined as

$$ \Omega_{i,j} = \frac{\vec d_{i,j}\cdot \vec E_0}{\hbar} $$

where $\vec d_{i,j}$ is the transition dipole moment for the $i\to j$ transition and $\vec E_{0}={\hat {\epsilon }}E_{0}$ is the vector electric field amplitude. $\hat\epsilon$ is the polarization of the light, which can be complex (e.g. for circular polarized light). But I think more importantly, $E_0$ is in general complex: $E_0 = \text{e}^{\text{i}\varphi}|E_0|$.

More information can be found in chapter 3.2.2 of this document (PDF link).

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