Extracting the uncertainty on a measured atomic transition

Hello! I have some measurements of a given transition in an atom, where each event consist of the measurement of this transition and the associated uncertainty (as details, the way it is done, is for each event recorded we measure the laser frequency and Doppler shift it to the frame of the atom), so my data is ##(f_i,df_i)##. For the purpose of this question we can assume that the uncertainty is the same for all measurements. From these measurements, I want to extract the central value of the transition and associated uncertainty. I am not sure how to do it properly. One way is to just take the mean of these values as the central value and the uncertainty would be ##\frac{df}{\sqrt{N}}##, where ##N## is the number of events. However, this doesn’t seem to account for the linewidth of the transition (I expect that the uncertainty on the central value to be smaller for a smaller linewidth). Another way to do it is to make a histogram of the measured frequencies, fit it with a Voigt (or Lorentz) profile and from there extract the central value, associated uncertainty and even the linewidth. However, in this case (when building a histogram), I am not sure how to include the uncertainty in the frequency when building the histogram. Basically, I am not sure how to account both for the uncertainty in individual measurements and the linewidth of the transition. Can someone help me with this? Thank you!

Are you attempting to determine the center frequency of the transition? The natural linewidth?

Are you certain that Doppler broadening is the only relevant broadening mechanism? If so, could you run the experiment Doppler-free?

I would think that simply fitting the observed line profile to the appropriate theoretical line profile should give you the answer. That should be straightforward with a Doppler-free spectrum.

Malamala said:

One way is to just take the mean of these values as the central value and the uncertainty would be dfN, where N is the number of events.

Yes, this is a decent method. This is the second-best way to estimate the error on the mean (based only on the assumption that your measurements are normally distributed).

Malamala said:

However, this doesn’t seem to account for the linewidth of the transition (I expect that the uncertainty on the central value to be smaller for a smaller linewidth).

Is ##df_i## is a measured or fitted uncertainty? By "fitted uncertainty" I mean you do your experiment once and get a spectrum (##\nu_n##,##y_n##) where ##\nu_n## is the probe frequency and ##y_n## is the spectroscopic data (for example: absorption, transmission, Ramsey contrast, etc). You then fit (##\nu_n##,##y_n##) to a gaussian/lorentzian/voigt profile and report the fitted center frequency and root-mean-square uncertainty on the fitted center frequency: (##f_i##,##df_i##). By "measured uncertainty" I mean do your experiment N times. Take the mean outcome and the standard error on the mean: (##f_i##,##df_i##).

If you meant ##df_i## is the measured uncertainty, then you don't have to do anything. Measured uncertainty is... the measured uncertainty. You can't "improve" it, since it already contains all the relevant information (like linewidth).

If you meant ##df_i## is the fitted uncertainty, then you just need to make sure your fitting routines (1) contain the linewidth as a fit parameter and (b) are returning good fits. If the fits look good to the eye, that's a good start. Next, I would check the covariance matrix (should be an option in whatever curve-fitting code you are using). If the covariance matrix is highly diagonal, you are good. If you see large off-diagonal terms in the covariance, then you fit is under-constrained and you need to add more data points (i.e., add more (##\nu_n##,##y_n##)'s). Once the fits look good, you should be able to interpret them as if they were measurement uncertainty.

Malamala said:

Another way to do it is to make a histogram of the measured frequencies, fit it with a Voigt (or Lorentz) profile and from there extract the central value, associated uncertainty and even the linewidth

I think you are confusing a transmission spectrum with a histogram. Your histogram of measured frequencies should not fit to a Lorentz or Voigt profile. Ideally, the histogram should fit to a Gaussian distribution, where the width is your measurement error. Also, crucially, the width of the histogram can be much smaller than the linewidth. Think about it: you can measure the center frequency of a transition with uncertainty less than the linewidth if you take enough data. In the real world, the histogram might deviate from a Gaussian because of drifts in the Doppler shift or other relevant experimental parameters.

The histogram method is the best way to find you measurement uncertainty overall.

Twigg said:

Yes, this is a decent method. This is the second-best way to estimate the error on the mean (based only on the assumption that your measurements are normally distributed).Is ##df_i## is a measured or fitted uncertainty? By "fitted uncertainty" I mean you do your experiment once and get a spectrum (##\nu_n##,##y_n##) where ##\nu_n## is the probe frequency and ##y_n## is the spectroscopic data (for example: absorption, transmission, Ramsey contrast, etc). You then fit (##\nu_n##,##y_n##) to a gaussian/lorentzian/voigt profile and report the fitted center frequency and root-mean-square uncertainty on the fitted center frequency: (##f_i##,##df_i##). By "measured uncertainty" I mean do your experiment N times. Take the mean outcome and the standard error on the mean: (##f_i##,##df_i##).

If you meant ##df_i## is the measured uncertainty, then you don't have to do anything. Measured uncertainty is... the measured uncertainty. You can't "improve" it, since it already contains all the relevant information (like linewidth).

If you meant ##df_i## is the fitted uncertainty, then you just need to make sure your fitting routines (1) contain the linewidth as a fit parameter and (b) are returning good fits. If the fits look good to the eye, that's a good start. Next, I would check the covariance matrix (should be an option in whatever curve-fitting code you are using). If the covariance matrix is highly diagonal, you are good. If you see large off-diagonal terms in the covariance, then you fit is under-constrained and you need to add more data points (i.e., add more (##\nu_n##,##y_n##)'s). Once the fits look good, you should be able to interpret them as if they were measurement uncertainty.I think you are confusing a transmission spectrum with a histogram. Your histogram of measured frequencies should not fit to a Lorentz or Voigt profile. Ideally, the histogram should fit to a Gaussian distribution, where the width is your measurement error. Also, crucially, the width of the histogram can be much smaller than the linewidth. Think about it: you can measure the center frequency of a transition with uncertainty less than the linewidth if you take enough data. In the real world, the histogram might deviate from a Gaussian because of drifts in the Doppler shift or other relevant experimental parameters.

The histogram method is the best way to find you measurement uncertainty overall.

Sorry I might not have been totally clear with my setup. For example, for one single atom, I know its energy, ##E_i## (I perform collinear resonant ionization spectroscopy) with a given uncertainty ##dE_i##. If I get one count (in this case I measure an ion due to ionization), by knowing the laser frequency (we can ignore the uncertainty on that) and the energy of the ion, I can doppler shift the laser frequency to the atom's rest frame, ##f_i##. This will have an uncertainty due to the error propagation of the energy, ##df_i##. So this is one event in my case, in which I record the frequency at which I get an event. Of course ##f_i## is not necessarily the central value of the transition of interest, given that the transition has a linewidth (also ##df_i## is not related to the linewidth, as it is given by the energy uncertainty alone). Now I repeat this N time, so I get N pairs of ##f_i## and ##df_i## for each atom. How do I extract the central frequency of the transition of interest (and ideally the linewidth) from this data? If I didn't have any uncertainty on individual frequencies i.e. ##df_i = 0##, I could just bin the data i.e. counts vs frequency, fit it with a Voigt profile and get the parameters of the transition from there. But I am not sure what to do when ##df_i## is not zero. Do I also just bin the data and ignore this uncertainty in my analysis (this doesn't seem right)?

Malamala said:

How do I extract the central frequency of the transition of interest (and ideally the linewidth) from this data?

Take the average of ##f_i## to get the center frequency. Some of your measurements will be red-detuned and some will be blue-detuned, but they should (to first order) occur with equal probabilities.

Malamala said:

Do I also just bin the data and ignore this uncertainty in my analysis (this doesn't seem right)?

This is exactly right. You're not ignoring the uncertainty in ##df_i##. Hopefully, ##df_i## is normally distributed about an average of 0. If this is the case, then it will be pulled out in the Voigt fit as a bias on the gaussian width, not the Lorentzian width. Check the fitting covariance between gaussian and loretzian widths to make sure the fit is able to distinguish them well.