Description

stuff=np.load(’sidebands.npz’) t=stuff[’time’] d=stuff[’signal’]
a) To start, model the data as a single Lorentzian and use analytic derivatives.Please use Newton’s method (or Levenberg-Marquardt if you prefer) to carry out the fit. What are your best-fit parameters for the amplitude, width, and center? Please parameterize the Lorentzian as

b) Estimate the noise in the data, and use that to estimate the errors in
your parameters.
c) Repeat part a), but use numerical derivatives. I suggest you use a helper function that accepts an input function (and any ancillary data/arguments you want to pass it) and returns the derivatives of that function with respect to the model parameters. Are your answers statistically significantly different from your answers in a)?
d) Repeat part c), but now model the data as the sum of three Lorentzians.The width of all three Lorentzians should be the same, and the separation of the side peaks from the main peak should be equal, i.e.:

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You should have sensible guesses for a,t0,w from earlier, so you should try to estimate reasonable initial guesses for b,c,dt. What are your errors on these parameters?
e) Look at the residuals from subtracting your best-fit model from the data.Do you believe the error bars you got by assuming the data are independent with uniform variance, and that the model is a complete description of the data?
f) Generate some some realizations for the parameter errors using the fullcovariance matrix ATN−1A from part d). Plot the models you get from adding these parameters to the parameter errors. What is the typical difference in χ2 for the perturbed parameters compared to the best-fit χ2? Is this reasonable?
h) The laser sidebands are separated from the main peak by 9 GHz (so dx maps to 9 GHz). What is the actual width of the cavity resonance, in GHz?
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