# Light pulses far from resonance

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

• 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. Nov 23 at 2:43