How to measure a 10 Hz energy splitting of two energy levels

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Hello! If I have 2 energy levels split by something of the order of 10 Hz (they can be connected by an electric dipole moment i.e. ##\Delta J = 0## and they have different parities), what would be the best way to measure this difference (even 10% error would be good, but the lower the error the better).

Here is what I have so far with my limited practical experience in spectroscopy: Doing normal Rabi/Ramsey spectroscopy I assume it wouldn't work, as I would need to generate a frequency at ~10Hz, which I don't think it's possible (or is it?). I can in principle still do a Rabi measurement with a detuned frequency, but I assume that the lowest frequency I can generate in practice would still be several orders of magnitude away from 10Hz, so it would take a very long time to populate the excited state enough to have good statistics, so I don't think that would work.

Another thing I can do is to use a Raman like measurement, where the 2 lasers have a frequency difference of 10Hz, but in practice lasers linewidths are much bigger than that, so I am not sure if I could get a good signal from there, i.e. if I can extract the central value with less than 10 Hz error (I don't know much about Raman type measurements, tho). I would appreciate any advice on this and comments on my ideas. Thank you!
 
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  • #2
Hello! If I have 2 energy levels split by something of the order of 10 Hz (they can be connected by an electric dipole moment i.e. ##\Delta J = 0## and they have different parities), what would be the best way to measure this difference (even 10% error would be good, but the lower the error the better). Here is what I have so far with my limited practical experience in spectroscopy: Doing normal Rabi/Ramsey spectroscopy I assume it wouldn't work, as I would need to generate a frequency at ~10Hz, which I don't think it's possible (or is it?). I can in principle still do a Rabi measurement with a detuned frequency, but I assume that the lowest frequency I can generate in practice would still be several orders of magnitude away from 10Hz, so it would take a very long time to populate the excited state enough to have good statistics, so I don't think that would work. Another thing I can do is to use a Raman like measurement, where the 2 lasers have a frequency difference of 10Hz, but in practice lasers linewidths are much bigger than that, so I am not sure if I could get a good signal from there, i.e. if I can extract the central value with less than 10 Hz error (I don't know much about Raman type measurements, tho). I would appreciate any advice on this and comments on my ideas. Thank you!
Greetings,

What kind of interaction is supposed to be giving rise to this 10Hz splitting?

Best regards,
ES
 
  • #3
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Greetings,

What kind of interaction is supposed to be giving rise to this 10Hz splitting?

Best regards,
ES
In principle that shouldn't matter too much, but for completeness we can assume that we have a magnetic fields bringing 2 Zeeman sublevels 10 Hz apart.
 
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  • #4
In principle that shouldn't matter too much, but for completeness we can assume that we have a magnetic fields bringing 2 Zeeman sublevels 10 Hz apart.
Greetings,

You specified two energy levels split by 10Hz, connected by an electric dipole allowed transition and having different parities. Of course the interaction matters, and I fail to see how a Zeeman field satisfies the conditions given.

If the two "split states" are both excited states, a quantum beat experiment might work in principle. But 10Hz is truly pressing it.

Now if you only care about being able to measure a spectral splitting of the order of 10Hz then the answer is that it is simplicity itself and done routinely every day. It is called nuclear magnetic resonance spectroscopy on diamagnetic organic molecules. Differences in chemical shifts and spin-spin couplings are trivially resolved at the level of 10Hz. But that is not the situation you specified.

Best regards,
ES
 
  • #5
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Greetings,

You specified two energy levels split by 10Hz, connected by an electric dipole allowed transition and having different parities. Of course the interaction matters, and I fail to see how a Zeeman field satisfies the conditions given.

If the two "split states" are both excited states, a quantum beat experiment might work in principle. But 10Hz is truly pressing it.

Now if you only care about being able to measure a spectral splitting of the order of 10Hz then the answer is that it is simplicity itself and done routinely every day. It is called nuclear magnetic resonance spectroscopy on diamagnetic organic molecules. Differences in chemical shifts and spin-spin couplings are trivially resolved at the level of 10Hz. But that is not the situation you specified.

Best regards,
ES
I am not sure I understand your answer (sorry). My questions is for paramagnetic molecules and the 2 levels are ground states. For example the 2 ground rotational levels in a molecule can be brought close together by using a magnetic field. But as I said, it can be by other means, too. For example 2 very close ##\Lambda##-doubling levels or even 2 levels that are so close by accident (as long as they are ground states or have very long lifetimes relative to the time of the experiment). All I am curious about is how to measure the difference between these 2 levels that somehow are very close together. Thank you!
 
  • #6
Twigg
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Since they are ground states, you have a chance. One path is to use two-color coherent Raman excitation to generate a pi/2 pulse (STIRAP would be ideal due to the speed, which bypasses the risk of decay to an undesirable state and loss of coherence). With your optical pi/2 pulse, you can do a Ramsey measurement. You won't be limited by laser linewidth the way you're thinking. The laser width gives you some phase error on the pi/2 pulse. This gives you an apparent frequency error of ##\sigma_\phi / T_2## where ## sigma_\phi## is the phase error due to laser linewidth and ##T_2## is the coherence time.

The state prep would be easiest if one of your two states was a stretched state (then you could prepare just the stretched state using ##\sigma^+## circularly polarized pumping light, then STIRAP a pi/2 pulse getting you on the equator of the Bloch sphere).

There's maybe a way to do coherent pi/2 pulses with microwaves and the nearest neighboring hyperfine or rotational level. I dont remember anything about doing coherent state transfer with microwaves,, it could be impossible for all I remember. I wish I'd paid attention to the routine details of quantum computing talks now lol. If it is possible, ion qubit folks will be doing it. I would look them up for you but I've got things this morning.

Another question: is the splitting 10Hz because the Bfield is tiny or is it because the Lande g-factor is tiny? If it's the first case, isolating that experiment from environmental fields will be hell. It'll go berserk anytime someone uses the elevator or the AC turns on. You'll want some mu metal shielding. You may also need to cryogenically cool the nearest radiating surfaces inside the vacuum chamber, depending on what the blackbody decoherence rate is like (10Hz should be fine but who knows?).
 
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  • #7
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Since they are ground states, you have a chance. One path is to use two-color coherent Raman excitation to generate a pi/2 pulse (STIRAP would be ideal due to the speed, which bypasses the risk of decay to an undesirable state and loss of coherence). With your optical pi/2 pulse, you can do a Ramsey measurement. You won't be limited by laser linewidth the way you're thinking. The laser width gives you some phase error on the pi/2 pulse. This gives you an apparent frequency error of ##\sigma_\phi / T_2## where ## sigma_\phi## is the phase error due to laser linewidth and ##T_2## is the coherence time.

The state prep would be easiest if one of your two states was a stretched state (then you could prepare just the stretched state using ##\sigma^+## circularly polarized pumping light, then STIRAP a pi/2 pulse getting you on the equator of the Bloch sphere).

There's maybe a way to do coherent pi/2 pulses with microwaves and the nearest neighboring hyperfine or rotational level. I dont remember anything about doing coherent state transfer with microwaves,, it could be impossible for all I remember. I wish I'd paid attention to the routine details of quantum computing talks now lol. If it is possible, ion qubit folks will be doing it. I would look them up for you but I've got things this morning.

Another question: is the splitting 10Hz because the Bfield is tiny or is it because the Lande g-factor is tiny? If it's the first case, isolating that experiment from environmental fields will be hell. It'll go berserk anytime someone uses the elevator or the AC turns on. You'll want some mu metal shielding. You may also need to cryogenically cool the nearest radiating surfaces inside the vacuum chamber, depending on what the blackbody decoherence rate is like (10Hz should be fine but who knows?).
Thanks a lot! This is really helpful! So in principle I can use STIRAP (this is what I meant by "Raman like measurement", sorry) and an excited electronic level to do this measurement? One thing I am not sure about is that excited state used in STIRAP (that doesn't get populated). I need to use 2 lasers connecting my 2 ground states (the ones 10 Hz apart) to this state, but these states have opposite parities, so won't it be impossible to connect them both to the same state (at least if I aim from E1 transitions)? Or is there a way to do that?

In the experiment I am thinking, the magnetic field would be large (of the order of 1 Tesla).
 
  • #8
Twigg
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In the experiment I am thinking, the magnetic field would be large (of the order of 1 Tesla).
That's a lotta oomph! :O

Ah classic STIRAP won't work for the reason you listed and because my dumb self didn't really think through the laser linewidth issue. You might get by with a trick. Phase lock two lasers to each other using a microwave source. Go up to excitated state using first laser, then shift to a different rotational line with the same microwave source (that gives you a third parity flip if you time it for a pi pulse), then second laser to go down. Since they all have well-defined relative phase, I think you'll get a STIRAP-like effect. You'll have to figure out the timings on your own, i recommend doing it numerically not analyticslly.

P.S. I remembered, you totally can do coherent transfers like pi pulses and ARP with microwaves.

OK, gotta run for real now lol
 
  • #9
f95toli
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One obvious question is how broad your levels are? And what type of broadening you expect?
Also, I am definitely not an optics expert (my background is more in area of ESR, i.e. I only ever use MWs) but do you really need two lasers for levels that are only 10 Hz apart? Wouldn't modulating one laser be more stable?
Or would that use up too much of the power?
 
  • #10
Twigg
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but do you really need two lasers for levels that are only 10 Hz apart?
In optics jargon, that's called coherent Raman transfer (if it's done with pi pulses instead of CW lasers, then it's called STIRAP). The issue @kelly0303 raises is that since the two ground states have opposite parity, Raman transfer doesn't have suitable transition rules. You need an odd number of dipole transitions to do this. If there's an allowed magnetic dipole transition, it's possible in theory but in practice you'd need the friggin death star for your laser. STIRAP is tough with electric dipole transitions because of the intensity requirements. That's why I was suggesting using two lasers and a microwave source, all with a coherent common-mode phase: it has the right parity rules and it has all the good things that STIRAP has (speed, efficiency, etc).

@f95toli, actually, your knowledge of microwaves techniques could be seriously handy in a situation like this. It might be easier to do this with a microwave source modulated by 10Hz. I just don't know if a microwave source with a <<10Hz linewidth exists.

Apparently, I have a problem stopping when I say I'm going to stop lol
 
  • #11
f95toli
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@f95toli, actually, your knowledge of microwaves techniques could be seriously handy in a situation like this. It might be easier to do this with a microwave source modulated by 10Hz. I just don't know if a microwave source with a <<10Hz linewidth exists.
Sure, 10 Hz is not very difficult. The short term stability should be much better (<1 Hz) than that for any generic microwave generator of the type you would use in the lab (generators from Keysight, R&S etc) .
However, if you want to do long measurements and/or need very good absolute frequency accuracy you probably want to either use an external Rb 10 MHz reference or a generator with an ovenized oscillator. Not because of the linewidth but because the generators can drift at the level of a few Hz over a period of hours if you are unlucky.
 
  • #12
Twigg
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I attached a little cartoon of what I had in mind for the 2 laser + 1 microwave STIRAP. @kelly0303, is the ground state structure I drew approximately what you have in mind? (I chose J = 0,1 and F = 1/2 for simplicity.)
 

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  • #13
Greetings,

Just my opinion, but I find the proposed experiment to be very poorly defined. A paramagnetic molecule in a 1Tesla field? How is the homogeneity of that external field controlled over the course of the experiment? Is that Paschen-Bach territory? If so, how does that effect the energy level structure and the applicable selection rules?

Does the molecule carry nuclear spin angular momentum? If so what are the hyperfine structure interactions and how do they respond to the Zeeman field? Why do the states separated by 10Hz not mix? The broadening questions mentioned above are also relevant.

Perhaps more to the point, the question appears to have changed dramatically between post #1 and post #5. I for one would appreciate a more definitive question from the outset.

Best regards,
ES
 
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  • #14
Twigg
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The field stability problem is brutal. Good catch ES137. @kelly0303, remember that the bohr magneton is about 1.4 MHz/G. That's 14 GHz/T. That means to get a stable 10Hz splitting with a 1T applied field, you need to stabilize the B field to on the order of 1 part per billion (ppb), or fluctuations under ##10 \mu G##. I haven't thought about this stuff in a while, but it sounds very difficult based on past experience.

Also good catch on Paschen-Bach regime. My cartoon should not contain F but instead ##\Omega## (assuming Hund's case (a)).

One possible trick to solve the field stability issue is co-magnetometry. Would need a lot more info to tell you if its feasible in your molecule. The ACME eEDM experiment has a comagnetometry state (i cant remember which state it was), so that might be a good place to look for inspiration.
 
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  • #15
My cartoon should not contain F but instead (assuming Hund's case (a)).
Nice arrows by the way :wink:
 
  • #16
Twigg
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I probably should've labeled those... but why spoil the mystery!
Red & purple: lasers connecting the ground states to different rotational levels in the excited state manifold, with polarization chosen to allow only one or the other of the two 10Hz split states
Blue: microwave connecting the excited state rotational levels
 
  • #17
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@Twigg @EigenState137 @f95toli thanks a lot for your answers. I am a bit confused about nuclear spin. I am sorry if I implied that, but my question has nothing to do with nuclear spin, the magnetic field interacts with the electron spin, hence why I mentioned a paramegnetic molecule. I am not sure I can answer all the questions about the actual experimental details, but here is a paper with something close with what I had in mind: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.120.142501

They apply a 0.5 T magnetic field in order to bring close to degeneracy 2 rotational states of the molecule. In that paper they search for parity violation, which means that there is an off-diagonal term that produces an avoided crossing, such that the minimum distance between the 2 levels is given by this term and let's assume it is of the order of 10 Hz. Assuming this setup (and say that in my case we can somehow keep the molecule in place, not have it move), my question is just what would be the way to measure this 10 Hz splitting.

@Twigg just to make sure I understand your idea, can I use 2 lasers and a microwave field, to drive a ##\pi## pulse between these 2 levels? And in principle instead of the 2 lasers I can use also microwave/RF fields (in order to couple some rotational levels in the same electronic state, instead of other electronic states) to do the same thing?
 
  • #18
Twigg
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Ah, ok, now I have an idea what you mean. I forgot about this parity violating physics search!

Since the PRL that was linked is behind an APS paywall, I'll summarize the relevant structure. The experiment is a search for a parity-violating, nuclear spin-dependent interaction between electrons and the nucleus in ##\mathrm{^{138}Ba^{19}F}##. The whole experiment takes place in the ground electronic manifold of BaF, which is the ##X^2 \Sigma## state and a Hund's case (b) state. As the OP says, the put a ~0.5T field to get into the Paschen-Bach regime where ##g_S## dominates over ##g_I##. Here's a recent paper about it on the arXiv.

For context, what @kelly0303 is proposing, an experiment like this with trapped molecules, hasn't been done to the best of my knowledge. Trapping is the key ingredient here, because molecular beam measurements have their precession time limited by the fact that the molecules are whizzing along through the finite experimental volume and literally smack into the other side of the vacuum chamber (longer flight path = longer interrogation time). Trapped molecules are hard, and that's why it hasn't been done. I vaguely remember Nick Hutzler is working towards a nuclear magnetic quadrupole moment (nuclear MQM) measurement in YbOH at Caltech, but I believe they decided against going for trapped YbOH, for reasons of Nick's tenure track and to avoid competition with ACME III. In the long run, there's nothing stopping the ACME III folks (who are going for ultracold YbOH in a dipole trap for eEDM measurements) from doing an MQM measurement with trapped YbOH, when they eventually get to that point. That's probably the better part of a decade off in the future before YbOH is ready, but who knows those groups have seriously surprised me before. Anyhow, that's the closest project that I know of to what @kelly0303 is describing.

@kelly0303 Yes, I was thinking a 3-pulse STIRAP sequence composed of two lasers and one microwave source, or 3 microwave sources. It's very possible that this is overkill. To do this, since as you say the laser linewidths will be greater than 10Hz, you will need to make for each laser pulse, you choose the polarization so one of the two parity states is forbidden. Since these are molecules, where transition rules are more like guidelines, you may want two degrees of forbidden-ness. If you use microwave sources, then as @f95toli says, you will have <<10Hz linewidths so you can be energetically forbidden. For STIRAP, if I remember correctly the trick is that all your driving fields need to have stable relative phases, and you need a fair bit of power.

As an aside, there are special tricks for molecular ions that I did not mention, but that's a whole 'nother ballgame.
 
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  • #20
Greetings,
If you use microwave sources, then as @f95toli says, you will have <<10Hz linewidths so you can be energetically forbidden.
You have suggested using pulsed excitation and that will certainly broaden the effective linewidth of the microwave source.

Thanks for the summary of the PRL paper.

Best regards,
ES
 
  • #21
Twigg
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You have suggested using pulsed excitation and that will certainly broaden the effective linewidth of the microwave source.
Hehe, I'm glad someone's around to call me on my goofs. :oldbiggrin: Good catch, once again! Maybe my first instinct, using the polarization to exclude one parity or the other, is the right only way to go.

I considered another option which would be to do this transfer at a different value of the applied magnetic field, say where the states are split by 1MHz, so you can gain the benefit of being spectrally resolved in addition to polarization resolved. You will then need to adiabatically ramp the magnetic fields to the final configuration at which the splitting is 10Hz. The problem is that to avoid Landau-Zener excitation, you'll have to go impossibly slow with the ramp, otherwise you'll see population in higher spin states of the ##X^2\Sigma## manifold.

Per LZ theory, $$\frac{dE}{dt} \approx g_S \mu_B \frac{dB}{dt}$$ where ##g_S \approx 2## and the probability of exciting the lower-energy parity state to the higher-energy parity state is given by $$P = e^{-2\pi \frac{V_{12}^2}{\hbar (dE/dt)}}$$ where ##V_{12}## is the width of the avoided crossing. Since ##V_{12}## is also on the order of ##2\pi \times 10\mathrm{Hz}##, we have $$P \approx e^{-(2\pi)^2 \frac{(10 \mathrm{Hz})^2}{2 \times (1.4 \mathrm{MHz/G}) \times (dB/dt)}} \approx e^{- \frac{3 \mathrm{mG/s}}{dB/dt}}$$ So, to get 1:1000 rate of Landau-Zener excitation, you'd need ##\frac{3 \mathrm{mG/s}}{dB/dt} = -\ln 10^{-3}## or ##\frac{dB}{dt} \leq 0.4 \mathrm{mG/s}##. Since you're ramping over roughly 1MHz to get down to 10Hz, that means you're ramping over a field strength of ##\frac{1 MHz}{2 \times (1.4 \mathrm{MHz/G})} \approx 0.35 \mathrm{G}##, and that means a total ramp duration of roughly ##\frac{0.35 \mathrm{G}}{0.4 \mathrm{mG/s}} \approx 15\mathrm{min}##. That's too much even for ions.
 
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  • #22
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Ah, ok, now I have an idea what you mean. I forgot about this parity violating physics search!

Since the PRL that was linked is behind an APS paywall, I'll summarize the relevant structure. The experiment is a search for a parity-violating, nuclear spin-dependent interaction between electrons and the nucleus in ##\mathrm{^{138}Ba^{19}F}##. The whole experiment takes place in the ground electronic manifold of BaF, which is the ##X^2 \Sigma## state and a Hund's case (b) state. As the OP says, the put a ~0.5T field to get into the Paschen-Bach regime where ##g_S## dominates over ##g_I##. Here's a recent paper about it on the arXiv.

For context, what @kelly0303 is proposing, an experiment like this with trapped molecules, hasn't been done to the best of my knowledge. Trapping is the key ingredient here, because molecular beam measurements have their precession time limited by the fact that the molecules are whizzing along through the finite experimental volume and literally smack into the other side of the vacuum chamber (longer flight path = longer interrogation time). Trapped molecules are hard, and that's why it hasn't been done. I vaguely remember Nick Hutzler is working towards a nuclear magnetic quadrupole moment (nuclear MQM) measurement in YbOH at Caltech, but I believe they decided against going for trapped YbOH, for reasons of Nick's tenure track and to avoid competition with ACME III. In the long run, there's nothing stopping the ACME III folks (who are going for ultracold YbOH in a dipole trap for eEDM measurements) from doing an MQM measurement with trapped YbOH, when they eventually get to that point. That's probably the better part of a decade off in the future before YbOH is ready, but who knows those groups have seriously surprised me before. Anyhow, that's the closest project that I know of to what @kelly0303 is describing.

@kelly0303 Yes, I was thinking a 3-pulse STIRAP sequence composed of two lasers and one microwave source, or 3 microwave sources. It's very possible that this is overkill. To do this, since as you say the laser linewidths will be greater than 10Hz, you will need to make for each laser pulse, you choose the polarization so one of the two parity states is forbidden. Since these are molecules, where transition rules are more like guidelines, you may want two degrees of forbidden-ness. If you use microwave sources, then as @f95toli says, you will have <<10Hz linewidths so you can be energetically forbidden. For STIRAP, if I remember correctly the trick is that all your driving fields need to have stable relative phases, and you need a fair bit of power.

As an aside, there are special tricks for molecular ions that I did not mention, but that's a whole 'nother ballgame.
Thanks a lot for this! Actually I would like to hear the tricks for molecular ions. As I said, the parity violation paper was just an example of what I was thinking. My questions is general i.e. if I give you a system with 2 ground state energy levels 10 Hz apart: molecule, atom, ion, due to magnetic/electric fields, accidental almost degeneracy, it doesn't matter, I am curious what are the best ways to measure that splitting.
 
  • #23
Greetings.
My questions is general i.e. if I give you a system with 2 ground state energy levels 10 Hz apart: molecule, atom, ion, due to magnetic/electric fields, accidental almost degeneracy, it doesn't matter, I am curious what are the best ways to measure that splitting.
This is not a rational approach to experimental physics. Start with the phenomenon of interest and then choose a good candidate system (atom, ion, molecule) that you expect to manifest that phenomenon. Then choose the experimental methods that you will attempt to apply. The experimental methodologies you ask about are highly complex, difficult to understand, and highly selective in terms of the phenomena and systems to which they are potentially applicable.

Again, you state that your question is general, that you wish to understand how to measure a splitting of 10Hz. That question was answered in post #4 of this thread.
Now if you only care about being able to measure a spectral splitting of the order of 10Hz then the answer is that it is simplicity itself and done routinely every day. It is called nuclear magnetic resonance spectroscopy on diamagnetic organic molecules. Differences in chemical shifts and spin-spin couplings are trivially resolved at the level of 10Hz. But that is not the situation you specified.
Alas, that was not to your liking because you were interested in a paramagnetic species. So which is it?

Best regards,
ES
 
  • #24
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Greetings.

This is not a rational approach to experimental physics. Start with the phenomenon of interest and then choose a good candidate system (atom, ion, molecule) that you expect to manifest that phenomenon. Then choose the experimental methods that you will attempt to apply. The experimental methodologies you ask about are highly complex, difficult to understand, and highly selective in terms of the phenomena and systems to which they are potentially applicable.

Again, you state that your question is general, that you wish to understand how to measure a splitting of 10Hz. That question was answered in post #4 of this thread.

Alas, that was not to your liking because you were interested in a paramagnetic species. So which is it?

Best regards,
ES
#4 didn't answer my questions at all... actually @Twigg's explanation did answer my question (by using STIRAP) and I am happy with that (hence why I asked him for more details). The question about ions is indeed extra, but if we are here and he is willing to tell me about that, I would be happy to hear.
 
  • #25
#4 didn't answer my questions at all
As noted earlier, inexplicit questions are difficult to answer.

At what point in this multi-thread series of questions does it become your burden to drag yourself to the library and undertake your own research?
 

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