Factor of 2 in the Rabi frequency This is a question about a strange literature discrepancy which I can't seem to resolve for myself.
In the wikipedia article on the Maxwell-Bloch equations, the Rabi frequency is defined as
$$\Omega = \frac{1}{\hbar}\underline{d}\cdot\underline{E}\,,$$
where $\underline{d}$ is the dipole moment of the transition. The density matrix equations of motion are then (only writing out for one of the elements and omitting the decay term):
$$\frac{d}{dt} \rho_{eg} = i \frac{\Omega}{2} (\rho_{gg} - \rho_{ee}) \,. $$
Essentially, my question is: Where does that divided by 2 come from?
Let me elaborate: The Hamiltonian in the $\underline{E}\cdot\underline{r}$ gauge within the dipole approximation should be $\underline{E}\cdot\underline{d}\,|e\rangle\langle g|+h.c.$, where the dipole moment is $\underline{d}=\langle g|\underline{r}|e\rangle$. Deriving the density matrix equation of motion with $\rho_{eg} = \langle e|\rho|g\rangle$ (and similarly for the other elements), I get the above equations but without the factor of $\frac{1}{2}$. Also the textbook by Scully&Zubairy seems to agree with me (Equation 5.3.24 in the 1997 Edition).
The wikipedia version seems to also be used elsewhere though, I have seen it in a couple of papers. So this issue is likely a difference in convention. However, I don't really see where a convention could enter here. The dipole moment is a pretty clear-cut quantity. One can have different conventions if one uses the Pauli $\sigma$-operators, but I have intentionally written the above in terms of states, which seem to allow no room for a convention.
Another option is that I am doing something terribly wrong, so any help is appreciated!
 A: It comes from the equation
$$
\cos(\omega t) = \frac{e^{i\omega t} + e^{-i\omega t}}{2}
$$
In the original equation you have
$$
E = E_0 \cos(\omega t)
$$
When you put the interaction Hamiltonian into Schrodinger's equation you get a term which in matrix notation is
$$
\left( \begin{array}{cc} 0 & E_0 \cos(\omega t)\\
E_0 \cos(\omega t) & 0 \end{array} \right)
$$
It is then common to adopt either an interaction picture or else in general a frame rotating at some angular frequency
$\omega_0$. Skipping the proof, the interaction term then becomes
$$
\left( \begin{array}{cc} 0 & e^{i \omega_0 t} E_0 \cos(\omega t)\\
e^{-i \omega_0 t} E_0 \cos(\omega t) & 0 \end{array} \right)
=
\frac{E_0}{2}
\left( \begin{array}{cc} 0 & 
e^{i (\omega_0+\omega) t}  +  e^{i (\omega_0-\omega) t}
\\
e^{i (-\omega_0+\omega) t}  +  e^{i (-\omega_0-\omega) t}
& 0 \end{array} \right)
$$
The terms involving $\omega_0 + \omega$ typically rotate very much more quickly than those involving $\omega_0 - \omega$. In that case we can make the rotating wave approximation which involves dropping the fast-evolving terms, leaving
$$
\frac{E_0}{2}
\left( \begin{array}{cc} 0 & 
 +  e^{i (\omega_0-\omega) t} \\
e^{i (-\omega_0+\omega) t}  & 0 \end{array} \right)
$$
The rotating-wave approximation is so widely used that people sometimes seem to think it is exact, and does not need to be mentioned. In fact it is not exact and a more thorough analysis reveals what is being left out. A quick rule of thumb is that it only gives a reasonable approximation when the Rabi flopping is itself slow compared to the other frequencies.
