Selection rule for spectra with circular polarization

[forever_physicist]

Hello everybody! I have a silly question that is blowing my mind.
When there is a circular polarized electric field, it can be interpreted as the real part of a complex field, for example
$$E(t) = E_0( \hat{x}+i\hat{y}) e^{-i\omega t}$$
Now, for some selection rules it is useful to calculate the matrix elements of the dipole operator in the direction of the electric field. If we use this definition that operator is
$$D_x + iD_y$$
while if we use directly the real electric field we get a different operator, that should be the correct one. Anyway, to derive the selection rules, usually this notation is used.
How can this work?

 

[vanhees71]

Usually one uses 1st-order perturbation theory, i.e., simply the matrix element of the dipole operator wrt. the unperturbed atomic states (Fermi's golden rule). So you can take the complex form and find the one for the real part by superposition.

 

[Twigg]

In most cases where the dipole operator is non-trivial, you will end up writing the dipole matrix in the molecule frame and the electric field in the lab frame. (You make them meet with the Wigner-Eckart theorem.) So don't bother trying to guess the right coordinates for $D$ in the lab frame at the start.