Undergrad Error on a transition frequency

[Malamala]

Hello! I am really new to this field, so I am sorry if my questions is silly or missing some parts. Please correct me if that is so. I am a bit confused about how well we can extract the value of a transition, say from ground to an excited state, of an atom (let's assume we can ignore any other energy levels, other than the probed ones i.e. 2 level system). Assume that we fit a Gaussian (it can be a Lorentzian or Voigt too) profile to our data (say we scan the frequency range with a laser) and from the fit we get a value for the center frequency of 100 and an error on the center (from the fitting program itself, which can be a chi-square minimization) of 3. Assume that the errors come only from the counting (i.e. Possion errors) and we have no systematics. Also assume that the sigma of the gaussian from the fit is 15 (due to Doppler thermal broadening). How well do we know the value of the transition? Would it be $100 \pm 15$ or $100 \pm 3$ or does it depends on other factors too? Also, assuming we reduce the temperature a lot and reach the natural width which is known to be (from theory or somehow else) 5, but we get an error on the fitting procedure of 3 again. Does it mean that we constrained the central value better than the natural linewidth? Is this even possible? So mainly, can someone explain to me how do we define the error on the measured transition given an experiment (and what other parameters that I miss we should take into account)? Thank you!

 

[BvU]

I am sorry

Stop apologizing ! (How are the other threads going ? Long forgotten ?)

Is this even possible?

Why not ? The top of a mountain can be located pretty accurately, even if the mountain is a few km wide !

If you get a distribution like (wikipedia) you certainly can pinpoint the peak position to within a small fraction of $\sigma$ !

 

[Malamala]

Stop apologizing ! (How are the other threads going ? Long forgotten ?)

Why not ? The top of a mountain can be located pretty accurately, even if the mountain is a few km wide !

If you get a distribution like (wikipedia) you certainly can pinpoint the peak position to within a small fraction of $\sigma$ !

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Thank you for your reply! I though, so, mathematically, however I am confused from a physics point of view. I know there are lots of efforts to reduce the thermal motion of atoms and molecules in experiments in order to measure their transitions frequencies. There are even specifically designed techniques meant to reduce the effect of doppler broadening (for example this). Also people are using narrow band laser to avoid broadening of the line coming from the laser itself. If one would be able to measure the central frequency even better than the natural linewidth, why would people try so hard to reduce the doppler effect, given that you can never get narrower than the natural linewidth anyway?

 

[BvU]

why would people try so hard to reduce the doppler effect

I can think of several reasons apart from determining the exact transition frequencies: fine and hyperfine structure and linewidth determination.
Perhaps someone with more experience can name a few other ones.

 

[Malamala]

I can think of several reasons apart from determining the exact transition frequencies: fine and hyperfine structure and linewidth determination.
Perhaps someone with more experience can name a few other ones.

That is true, but I was asking in the more simplified case of having just 2 levels (no other complications coming from the structure of the atom itself). In the link I sent you, this is what they assume, so I guess there is a limitation even when fine/hyperfine structure, interaction with other levels etc. are not taken into account. I am just not sure what sets these limitations. Or am I missunderstaning that wikipedia article?

 

[f95toli]

There are at least three answers to your question

1)The main reason for why people want to get better at measuring linewidths is that we want to build better clocks. Current cesium clocks have a stability of 1 part in 10^15 or so, but optical clocks (which have a narrower linewidh because they use optical instead of microwave transitions) can to to 1 part in 10^18 or even somewhat better. It is not at all obvious which system is best for this so people are looking at a variety of atoms and ions in several different configurations.

2) Measuring the "natural" linewidth is not trivial in real life. In many cases the linewidth you measure is limited by "technical" (Doppler shifts etc) effects which in principle can be removed. In most (but not all) atomic clocks you are also not measuring a single ion but a dilute cloud, this means that different ions will experience slightly different magnetic field and laser intensities and this also leads to broadening of your measurement.

3) There is no such thing as a single value for the measured frequency. In real life you need to measure for some time to reduce the error (from "white" noise) and if you plot the error of the measured frequency as a function of measurement time yo will see that that it first goes down, but then flattens out and eventually start going up again. The latter is causes by systematic drift in your measurements.
This is why results from clocks are always presented as an Allan variance/deviation
See
https://en.wikipedia.org/wiki/Allan_variance#/media/File:AllanDeviationExample.gif

A lot of the work that goes into this is related to removing the systematic errors and the 1/f noise (which is the flat part of the curve). Note that the drift part is not part of the normal distribution; if you measure for long enough you will find that most distributions are skewed because of various drift mechanisms.