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I've been doing some reading and came across a simple implementation of the Hadamard gate using Rabi oscillations of an atom in a laser field. However, the author mentions that it required the Rabi frequency $\Omega$ to be taken as purely imaginary.

I know that $\Omega$ is defined in terms of the field amplitude which is in general complex, so on the face of it this seems okay. But I was always taught that in optics while we work with complex field amplitudes and such, it is only the real component that represents the actual, physical field. And we introduce complex numbers to make the algebra easier. After all, the electric field is never complex when you measure it.

How can the Rabi frequency, which is the measurable frequency of something undergoing Rabi oscillations be complex? What is the physical interpretation of such a thing? I don't see how we could tune the laser field to make a complex Rabi frequency because frequencies can never be complex, how can we implement a gate like this?

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Maybe the confusion comes from the fact that we use the name "Rabi frequency" for two slightly different things: the typical frequency of Rabi oscillations (which is a real number) and the amplitude appearing in the effective Hamiltonian in the rotating wave approximation, associated to atomic transitions like $|g\rangle\to |e\rangle$ (which can be complex). The former can be regarded as the modulus of the latter.

More quantitatively, the light-atom interaction Hamiltonian, in the dipole approximation, is $\hat{H}_{\mathrm{int}} = - \hat{\pmb{d}}\cdot \pmb{E}$, where $$ \hat{\pmb{d}} = \pmb{d}_{ge} |g\rangle \langle e | + \pmb{d}_{ge}^* |e\rangle \langle g| $$ is the dipole moment operator of the atom and $$ \pmb{E}(\pmb{r},t) = E_0 (e^{i(\pmb{k}\cdot\pmb{r}-\omega t)} \pmb{u} + \mathrm{h.c.} ) $$ is the electric field of amplitude $E_0$, frequency $\omega$, polarization $\pmb{u}$ (in Jones' notation) and wavevector $\pmb{k}$. The non interacting Hamiltonian of the atom is instead $\hat{H}_0 = \hbar \omega_0 |e\rangle \langle e|$.

The effective Hamiltonian in the rotating wave approximation turns out to be $$ \hat{H}_{\mathrm{RWA}} = \left( \begin{array}{cc} 0 & E_0 \pmb{d}_{ge}\cdot \pmb{u}^*\, e^{i(\pmb{k}\cdot\pmb{r} - \omega t)} \\ E_0 \pmb{d}_{ge}^* \cdot \pmb{u}\, e^{-i(\pmb{k}\cdot\pmb{r} - \omega t)} & 0 \end{array} \right). $$ The complex Rabi frequency $\Omega$ is defined as $$ \hbar \Omega = E_0 \pmb{d}_{ge}\cdot \pmb{u}^* e^{i\pmb{k}\cdot\pmb{r}} $$

Hope this helps!

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