Magnetic Resonance — Question 4

Question 4

Detuning limits the maximum transfer even for an ideal pulse

For a spin initially in its lower state, use $P_e(t)=\frac{\Omega^2}{\Omega^2+\delta^2}\sin^2[\sqrt{\Omega^2+\delta^2}\,t/2]$, with angular detuning $\delta=\omega-\omega_0$. Let $\delta=\sqrt3\Omega$.

1. Find the largest achievable excitation probability.

2. Find the first time this maximum occurs and compare it with the resonant inversion time.

Question 4 - Solution

Detuning limits the maximum transfer even for an ideal pulse

(a) Method and calculation

$$\Omega_{\mathrm{eff}}=\sqrt{\Omega^2+3\Omega^2}=2\Omega.$$$$\boxed{P_{e,\max}=\Omega^2/(4\Omega^2)=1/4}.$$

(b) Interpretation and further calculation

$$\Omega_{\mathrm{eff}}t_{\max}/2=\pi/2\Rightarrow\boxed{t_{\max}=\pi/(2\Omega)}.$$$$t_\pi=\pi/\Omega\Rightarrow\boxed{t_{\max}=t_\pi/2}.$$

Physical check

Faster oscillation away from resonance does not imply more efficient inversion. The rotation axis tilts and the accessible population excursion decreases.