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!


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