NormalizedIntensity Amplitude
Fit
(a)
−20 −10 0 10 20 Frequency / MHz - 975 734 900 0.0
0.5 1.0
NormalizedIntensity Amplitude
Fit
(b)
Figure 10.2:(a) Fluorescence spectrum of theN0= 1,J0= 3/2,F0= 1/2←f,F00= 1/2, 3/2 transition cluster in OD, with full quantum mechanical fit. The close spacing between the individual transitions complicates the fit. (b) In contrast, the fluorescence intensity of theN0 = 1,J0 = 3/2,F0 = 3/2←f,F00 = 1/2, 3/2 and 5/2 transition cluster of OD shows higher separation of the individual transitions.
a more detailed analysis impossible. In the OH measurements and measurements of OD transitions with a single hyperfine component, the p3 parameter is constrained to fall in the 0.2-0.4 range determined in the OD measurements. This is necessary, since these spectra only offer a single peak, and the information needed to determine the laser power is insufficient.
10.2 Uncertainty and Simple Voigt Fit
The fitting procedure of the full QM fit supplies absolute, zero-field positions of the OH and the OD transitions. However, each fitted line position has an associated uncertainty.
The task of the simple Voigt model is to assign a line position excluding the AC Stark shift and the Zeeman effect in order to determine the contribution of these effects to the total measurement uncertainty.
The total uncertainty is determined in multiple steps. Each transition measurement is repeated at least twice on different days, sometimes with different laser powers. Fitting a single fluorescence spectrum results in an estimate of a transition frequency νi with a statistical uncertainty σi. Including the uncorrelated uncertainty of 10 kHz, which comes from the fluctuations of the retroreflected beam (Section 9.8.1), increases the uncertainty to
σi02 =σi2+ (10 kHz)2. (10.2)
10.2. Uncertainty and Simple Voigt Fit Now each single transition measurement is associated with an uncertainty of σi0, which supplies a weighting factor for the calculation of the weighted mean
¯ The uncertainty of the final transition frequency ¯ν is estimated with σν¯. A particular transition frequency is determined from N individual transition measurements, with the sum computed over all of those. However, a few very small individual uncertaintiesσi0 leads to an underestimate of the errorσν¯ relative to the spread of the individual frequencies νi. In contrast, calculating the standard deviation of the unweighted mean results in a larger uncertainty of ¯ν. In the end, we choose the larger of the two values as the total statistical uncertainty
The estimated uncertainties are in general of the same order of magnitude as the Zee-man effect or the AC Stark shift. Frequently, the effects of such shifts are estimated by varying the magnetic field or laser intensity in the measurements, but the large relative uncertainties make extracting their contributions in this manner unfeasible.
The simple Voigt spectrum aims to detect changes of the line positions, by fitting the full QM model (Section 10.1). Since the full QM model supplies the zero-field line positions, the simple model estimates the field dependent line positions. The fitting function is based on a sum of Voigt profiles with the Lorentzian width Γ, Gaussian width ˜p4, transition frequenciesνi and correspond-ing transition dipole moments µi. The free parameters are ˜p0-˜p4, where the first two parameters ˜p0 and ˜p1 are a vertical scaling and offset that adjust the simple model rel-ative to the simulated spectrum of the full QM model. Saturation effects that result in peak strengths in the simulated OD spectra that do not match the expected µ2i scaling are accounted with the empirical saturation parameter ˜p2. In spectra with a single peak where there is insufficient information to determine ˜p2, the prefactor ˜p2h
1−exp−µ2 i
˜ p2
i is replaced with µ2i. In analogy to the full QM model, the parameter ˜p3 accounts for the line positions. Based on ˜p3, the contributions of the Zeeman effect and the AC Stark shift are assigned.
The full QM model I1(ν) supplies a parameter set of its own p0-p4. After fitting
Chapter 10. Analysis
the model to the measured spectrum, those parameters are well defined. Additionally, the effective Hamiltonian H0 used in this fit is based on the ambient magnetic field of B = 75µT. With these two constraints, the modelI1(ν) describes a fluorescence spectrum separated from noise (Figure 10.3). Changing the magnetic field input of the effective
Figure 10.3:Schema of the simple multi-Voigt fit procedure. Based on the previously fit of the full QM model I1(ν) the parametersp0-p4are fixed. However, changing the magnetic field strength inside the effective Hamiltonian generates slightly different spectra. The simple multi-Voigt modelI2(ν) assigns a net shift to these variations after a fit to the simulated spectrum.
Hamiltonian results in slightly different fluorescence spectra I1(ν). The fit of the simple multi-Voigt model I2(ν) to the simulated spectrum of I1(ν) allows an estimate of the transition frequency net shift. The parameter ˜p3 determines the new line positions, by subtracting it from the zero-field transition frequencies computed with I1(ν). Based on the magnetic field measurement uncertainty of ±5µT, the magnetic field might well be 70µT or 80µT. Thus, the fit of the simple modelI2(ν) to the full QM modelI1(ν) supplies new line positions atνVoigt,80µT andνVoigt,70µT, based on the two different magnetic fields.
We estimate the uncertainty of the transition frequency due to the Zeeman effect to be
∆νZeeman= νVoigt,80µT−νVoigt,70µT
2 . (10.6)
In contrast to the magnetic field dependence of the Zeeman effect, the AC stark shift depends on the laser power. However, the full QM model shows a discrepancy between the measured powers and the fitted values, which are significantly less than half of the measured values. Thus, we estimate the uncertainty due to the AC Stark shift and saturation effects based on the difference
∆νStark =νQM,75µT−νVoigt,75µT, (10.7)
with the fitted transition frequencyνQM,75µT of the full QM model I1(ν) and the modified line positionνVoigt,75µT of the simple Voigt model I2(ν). In the following, we assume the laser power discrepancy and the ambient magnetic field are constant over the course of all measurements, which means the corresponding uncertainties are correlated. Therefore, the
10.3. Zero-field Line Positions