Ramsey spectroscopy and spin echoes

[Malamala]

Hello! Assuming we use a laser of frequency very close to resonance, in the Ramsey technique (say for 2 level atoms) the $\pi/2$ pulse would put the Bloch vector in the equatorial plane, along the y axis, then in the free region the vector will rotate around the z axis accumulating a phase of $\Delta T$, where $T$ is the free evolution time and $\Delta$ is the detuning. After that, a second $\pi/2$ pulse followed by a readout will give us Ramsey fringes (here I am ignoring the lifetime of the excited state). However if we have some unhomogeneous broadening, during the free evolution time, different atoms in the ensemble will rotate at different frequencies, so in the end the signal will be significantly reduced. I read that spin echo like techniques can solve this, but I am not sure I understand how. In spin echo, in the middle of the free evolution time you apply a $\pi$ pulse, such that by the end of the evolution time, any effect of different rotation frequencies is cancelled. However, in that case you end up with the Bloch vector pointing always in the $-y$ direction (assuming the first $\pi/2$ pulse placed the vector along the $+y$) and it seems like this happens no matter what the detuning is (assuming is small enough), simply because you do a mirror reflection of the motion half way through. So you lose any information about the detuning. What am I missing here. How can you still get Ramsey fringes when using this spin echo technique? Thank you!

 

[Fred Wright]

If you isolate your experiment such that the only cause of detuning is 1/f noise, then you can take advantage of the fact that with 1/f noise there is no energy transfer to the environment. 1/f noise is reversible to some extent. Apply the $\frac{\pi}{2}$ pulse to the system with the spin vector aligned with the z axis of the Bloch sphere. The spin vector is knocked to x-y plane and suffers free evolution with its phase being slightly jostled about by the 1/f noise and detuning for a time $T_2$. Then the $\pi$ pulse is applied which reflects the spin vector in the x-y plane. The system again suffers free evolution for time $T_2$ but the phase trajectory of the spin vector doubles back on itself cancelling the first detuning period (Amazing!). A second $\frac{\pi}{2}$ is applied kicking the spin vector back to original orientation along the z axis.

 

[Malamala]

If you isolate your experiment such that the only cause of detuning is 1/f noise, then you can take advantage of the fact that with 1/f noise there is no energy transfer to the environment. 1/f noise is reversible to some extent. Apply the $\frac{\pi}{2}$ pulse to the system with the spin vector aligned with the z axis of the Bloch sphere. The spin vector is knocked to x-y plane and suffers free evolution with its phase being slightly jostled about by the 1/f noise and detuning for a time $T_2$. Then the $\pi$ pulse is applied which reflects the spin vector in the x-y plane. The system again suffers free evolution for time $T_2$ but the phase trajectory of the spin vector doubles back on itself cancelling the first detuning period (Amazing!). A second $\frac{\pi}{2}$ is applied kicking the spin vector back to original orientation along the z axis.

But I am not sure I understand how that helps. In a Ramsey technique, assuming we have no noise, during the free evolution time, the vector rotates in the x-y plane, and the second $\pi/2$ pulse leads to a projection back on the z axis. However, the idea (as far as I understand) is that this projection will lead to some of the spins pointing upwards, some downwards i.e. you don't get 100% to the original state. And it is based on this amount of up versus down that you gain the information about the detuning. However in the spin echo technique, you end up with all the spins pointing in the same direction as in the beginning, regardless of the detuning. So I am not sure how do I extract the detuning anymore in that situation. It seems like that information is lost.

 

[f95toli]

Maybe I don't understand the question correctly, but in normal Hahn-type sequence (pi/2-pi/pi/2 you can't extract the detuning.
The typical result of a Hahn echo experiment will be exponentially decaying curve(where each point is e.g. the pulse area. The curve decays with a time constant T2; i.e. the coherence time of the system. T2 is usually what you are trying to measure with an echo experiment.
If you have an ideal two-level system an exponential decay should be all you see if you've tuned up your pulses properly (which is an important if)

In many real systems the results are more complicated. You can e.g. have multiple time constants and in e.g. ESR you can also get modulations (oscillations) due to the interaction the the nucleus

There are also more complicated sequences and I believe (I've never used them) there are some examples of Ramsey-type experiments for three level systems which do involve a one or more pi pulses (more generally a full CPMG sequence) but the pulses are then applied at different frequencies.