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I have a doubt when defining pulses in optics, as $\pi$ or $\pi/2$ pulses for systems of energy levels.

Theoretically, we have that in a 2-level atom in the ground state,

  • $0$ Pulse $\qquad t=0 \quad \; \ \ \xrightarrow{} \quad |\psi\rangle=|\tilde{1}\rangle$

  • $\pi/2$ Pulse $\quad t=\frac{\pi}{2\Omega} \quad \xrightarrow{} \quad |\psi\rangle=\frac{1}{\sqrt{2}}(|\tilde{1}\rangle+|\tilde{2}\rangle)$

  • $\pi$ Pulse $\qquad t=\frac{\pi}{\Omega} \ \quad \xrightarrow{} \quad |\psi\rangle=|\tilde{2}\rangle$

That can be seen as a rotations in the Bloch optic sphere arround the precession vector $$\begin{pmatrix}-\Omega \\ 0\\ \Delta\end{pmatrix}$$

So it's easy to see that in the case of resonance, the evolution of the system is given by rotations around $x$ axis $ \begin{pmatrix}1&0&0\end{pmatrix}$, but in the case of far resonance $\Delta \gg \omega$ there are almost rotations at the $z$ axis $ \begin{pmatrix}0&0&1\end{pmatrix}$.

For example if you have the case with the initial state in the ground state $ \begin{pmatrix}0&0&-1\end{pmatrix}$, when you apply the rotations the state will not change, but will change the phase, in this case,

  • $0$ Pulse $\qquad t=0 \quad \ \ \ \ \xrightarrow{} \quad |\psi\rangle=|\tilde{1}\rangle$

  • $\pi/2$ Pulse $\quad t=\frac{\pi}{2(?)} \quad \xrightarrow{} |\psi\rangle=-i|\tilde{1}\rangle$

  • $\pi$ Pulse $\quad \ \ \ t=\frac{\pi}{(?)} \quad \ \xrightarrow{} |\psi\rangle=-|\tilde{1}\rangle$

  • $3\pi/2$ Pulse $ \ t=\frac{3\pi}{2(?)} \quad \xrightarrow{} |\psi\rangle=i|\tilde{1}\rangle$

My question is what frequency you have to put in the place that i left as $(?)$, the Rabi frequency $\Omega$ or the generalised Rabi frequency $\Omega'=\sqrt{\Delta^2+\Omega^2}$? because is not the same, if we are far from resonance we have, $$\Omega'\approx \Delta \left( 1+\frac{\Omega^2}{2\Delta^2}\right)$$

And I don't know how to derive this result to see if is one or the other

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  • $\begingroup$ When you say the rotation is about $(0,0,1)$ on the Bloch sphere, you're making a "zeroth-order" approximation, meaning the field is infinitely far detuned. It would be like not having a field at all, so the Rabi frequency being ill-defined makes sense. But your $\Omega'$ equation is a first-order approximation. That accounts for the difference in mathematical behavior you're seeing. $\endgroup$ Nov 23 at 2:43

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