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?

• I also found a mistake on Eberly's book... he should use k/2 on a step and he uses k. Correcting this I can get the correct optimal amplitude. – Igor César De Almeida Sep 6 '18 at 12:32

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: $$$$c_0(t) = \cos(\Omega t) \\ c_1(t) = i \sin(\Omega t)$$$$ In this case, $\Omega$ describes the frequency at which the amplitudes of the wavefunction oscillate.
However, commonly one is interested in the oscillation in the populations in the states $|0\rangle$ and $|1\rangle$. The populations are defined by the magnitude squared of the amplitudes: $$$$|c_0(t)|^2 = \cos^2(\Omega t) = \frac{1}{2} + \frac{1}{2}\cos(2\Omega t)\\ |c_1(t)|^2 = \sin^2(\Omega t) = \frac{1}{2} - \frac{1}{2}\cos(2\Omega t)$$$$ 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.