Two different expressions for Rabi Frequency (on resonance) I've been searching through several articles and books and I found two different expressions for the Rabi Frequency on the semiclassical rotating-wave approximation, and they are different by a factor of two. 
Consider a eletric field $$\boldsymbol{E}=E_0\sin(\omega t)$$ and a dipole matrix $$\left(\begin{array}{cc}
0 & \mu\\
\mu & 0
\end{array}\right)$$
In some places (when people derive it using Heisenberg picture), the formula is:
$$E_0=\frac{\pi\hbar}{2\left\langle a\right|\mu\left|b\right\rangle t}$$
(like on Allen,Eberly book called "Optical Ressonance and Two Level Atoms", or these notes by Oliver Benson, eq 293)
In other places (like Wikipedia or the paper arXiv:0707.1883 at eq A.18) the formula is:
$$E_0=\frac{\pi\hbar}{\left\langle a\right|\mu\left|b\right\rangle t}.$$
I've been through both derivations step by step and I could not understand why they are different. Am I misunderstanding the meaning of something on any of the two formulas?
 A: Unfortunately, there are two different definitions of Rabi frequency that differ by a factor of two. Consider a qubit undergoing Rabi oscillations: it has a time-dependent wavefunction given by $|\psi(t)\rangle = c_0(t) |0\rangle + c_1(t)|1 \rangle$. One definition of the Rabi frequency would be to write the time-dependent coefficients as:
$$
\begin{equation}
c_0(t) = \cos(\Omega t) \\
c_1(t) = i \sin(\Omega t)
\end{equation}
$$
In this case, $\Omega$ describes the frequency at which the amplitudes of the wavefunction oscillate.
However, commonly one is interested in the oscillation in the populations in the states $|0\rangle$ and $|1\rangle$. The populations are defined by the magnitude squared of the amplitudes:
$$
\begin{equation}
|c_0(t)|^2 = \cos^2(\Omega t) = \frac{1}{2} + \frac{1}{2}\cos(2\Omega t)\\
|c_1(t)|^2 = \sin^2(\Omega t) = \frac{1}{2} - \frac{1}{2}\cos(2\Omega t)
\end{equation}
$$
Note that here, the oscillation in the populations actually occurs at twice the frequency; this $2\Omega$ is often used as the definition of Rabi frequency instead.
