# Ramsey spectroscopy and spin echoes

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• Malamala
In summary, the spin echo technique can solve the issue of unhomogeneous broadening in Ramsey experiments by using 1/f noise, which is reversible and does not transfer energy to the environment. By applying a pi/2 pulse to align the spin vector with the z axis, and then a pi pulse in the middle of free evolution time, the spin vector's phase trajectory doubles back on itself and cancels out the first detuning period. This allows for a second pi/2 pulse to be applied, resulting in all spins pointing in the same direction as the beginning, regardless of the detuning. However, this technique does not allow for the extraction of detuning information, as it only measures the coherence time of the system.
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!

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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.

Fred Wright said:
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.

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.

## 1. What is Ramsey spectroscopy?

Ramsey spectroscopy is a technique used in atomic and molecular physics to measure the frequency of a specific transition between energy levels. It involves using two laser pulses to manipulate and measure the quantum state of a system.

## 2. How does Ramsey spectroscopy work?

In Ramsey spectroscopy, the first laser pulse is used to prepare the system into a superposition of two energy states. The system then evolves for a specific time before a second laser pulse is applied to measure the population of the two states. The difference in the measured population gives the frequency of the transition between the two states.

## 3. What is the significance of spin echoes in Ramsey spectroscopy?

Spin echoes are used in Ramsey spectroscopy to improve the precision and accuracy of the frequency measurement. They are created when a third pulse is applied to the system at a specific time, causing the two energy states to rephase and cancel out any external magnetic field fluctuations that could affect the measurement.

## 4. What are the advantages of using Ramsey spectroscopy over other techniques?

Ramsey spectroscopy is a non-destructive technique, meaning that the system is not altered or destroyed during the measurement. It also has high sensitivity and precision, making it useful for studying small energy differences between states.

## 5. What are some real-world applications of Ramsey spectroscopy?

Ramsey spectroscopy has many applications in fields such as atomic clocks, quantum computing, and precision measurements. It is also used in studying fundamental physics, such as testing the validity of the laws of quantum mechanics.

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