Open the preparation paper before drawing the line
Open Pereira Dos Santos and colleagues’ 2002 author manuscript, arXiv:physics/0209010v1. Follow the preparation in Figures 1–2, the ratio test in Figure 3, then Tables I–II. The problem is that more atoms improve counting precision while their interactions also shift the clock frequency. The experiment changes the density, measures the change, and estimates what frequency would remain at zero density.
Effective density means the density sampled by the interacting atoms during the measurement. The detector directly counts atoms that arrive later. Relating those two quantities is the practical work behind the extrapolation.
Read the evidence: F. Pereira Dos Santos et al. (2002), Controlling the cold collision shift in high precision atomic interferometry — arXiv:physics/0209010v1S. Bize et al. (2005), Cold atom Clocks and Applications — author manuscript, clock operation and collision control
Track which atoms survive the selection
In Figure 1, the microwave amplitude and frequency are shaped during a passage through resonance. The complete pulse transfers atoms to the selected internal state. Stopping at the midpoint prepares equal weights of the two states, after which a light pulse removes the unwanted state. This is the intended way to halve the sample while preserving its shape and velocity distribution.
Figure 2 compares the transfer probability with the drive strength. Identify measured points separately from the calculated curves and calculated points for the cavity field. The useful region is where full and half transfers are insensitive to small changes in drive.
Then inspect Figure 3. It tests how steadily the measured number ratio can be reproduced. The reported mean is near 0.506. Keep that measured value and its calibration uncertainty when estimating a correction; substituting an exact half would be a further assumption.
Read the evidence: F. Pereira Dos Santos et al. (2002), Controlling the cold collision shift in high precision atomic interferometry — arXiv:physics/0209010v1
Calculate the intercept with a known ratio
Assume frequency varies linearly with effective density and both preparations share the same collision coefficient. The subscripts denote the low and high settings. The ratio is the low effective density divided by the high effective density, strictly between zero and one. Eliminating the common slope gives the zero-density frequency below.
For a constructed example, suppose the frequency offsets from an arbitrary reference are plus 0.4 millihertz at high density and plus 0.2 millihertz at half that density. Extending the line gives zero offset. Keeping the same frequencies but using a ratio of 0.55 instead gives about minus 0.044 millihertz. A small calibration error has moved the extrapolated answer.
Read the evidence: F. Pereira Dos Santos et al. (2002), Controlling the cold collision shift in high precision atomic interferometry — arXiv:physics/0209010v1
Check that the two preparations really share a slope
Changing loading time can change the cloud size or temperature as well as atom number. An ordinary selection pulse in an uneven microwave field can favour certain trajectories. In either case, detected atom counts can stop being proportional to the density that matters for collisions.
Tables I–II compare temperatures and preparation methods. Use the table headings to distinguish the measured frequency difference, number ratio and inferred coefficient. Table I lists statistical and systematic uncertainty separately for the corrected-frequency comparison. Table II tests whether changing the selection method changes the inferred collision coefficient.
A useful extra check is to measure more than two effective densities under a controlled preparation. Two points always define a line; a third can reveal curvature. A wrong density calibration can still shift a straight-looking line, which is why preparation checks and the fit need to be read together.
Read the evidence: F. Pereira Dos Santos et al. (2002), Controlling the cold collision shift in high precision atomic interferometry — arXiv:physics/0209010v1
Carry the uncertainty through the extrapolation
At a density ratio of one half, the intercept equals twice the low-density frequency minus the high-density frequency. If the two frequency estimates are independent with equal standard uncertainty, the intercept uncertainty is larger by the square root of five. Shared oscillator noise adds covariance, and uncertainty in the density ratio supplies another contribution.
Nearly equal densities give little leverage: the small denominator amplifies measurement errors. For the next correction paper, retain the two frequencies, the density-ratio calibration, the preparation protocol, the linearity checks and the uncertainty calculation. The final zero-density frequency is an extrapolation from measurements performed with real, finite samples.
Read the evidence: F. Pereira Dos Santos et al. (2002), Controlling the cold collision shift in high precision atomic interferometry — arXiv:physics/0209010v1S. Bize et al. (2005), Cold atom Clocks and Applications — author manuscript, clock operation and collision control