More atoms change two things
More independent detected atoms can reduce population-counting uncertainty. In a cold cesium fountain, interactions between atoms can also shift the measured frequency. Improving counting statistics therefore creates an experimental tradeoff if density-dependent corrections are not known well enough.
Pereira Dos Santos and colleagues addressed this problem in 2002 by controlling population transfer before Ramsey interrogation. The key was to change the number of atoms while preserving the spatial and velocity distributions closely enough that the ratio of effective collision densities could be known.
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
Write the extrapolation assumption explicitly
Assume the measured frequency depends linearly on effective density with one common coefficient. Label the low- and high-density frequencies accordingly, and let the ratio be the low effective density divided by the high one. Eliminating the common coefficient gives the zero-density intercept below.
The ratio lies strictly between zero and one. It is an effective collision-density ratio, not automatically the ratio of fluorescence counts. To use a detected-number ratio instead, preparation and detection must preserve the relevant relation between counted atoms and the density sampled during interrogation.
Read the evidence: F. Pereira Dos Santos et al. (2002), Controlling the cold collision shift in high precision atomic interferometry — arXiv:physics/0209010v1
An incorrect ratio moves the intercept
Choose invented frequency offsets relative to an arbitrary reference: plus 0.4 millihertz at high density and plus 0.2 millihertz at half that density. The linear extrapolation returns zero offset. The cancellation can be checked directly from the two underlying density points.
Now keep those same measured offsets but mistakenly use a density ratio of 0.55. The inferred intercept becomes about minus 0.044 millihertz. Taking more data at the same two preparations can reduce their counting scatter while leaving this ratio error in the answer.
Read the evidence: F. Pereira Dos Santos et al. (2002), Controlling the cold collision shift in high precision atomic interferometry — arXiv:physics/0209010v1
Half a population need not mean half the density
Changing loading time can alter cloud size and temperature as well as atom number. Changing an ordinary selection-pulse area can weight trajectories differently if the microwave field is nonuniform. Two samples with a simple detected-count ratio may then have different collision coefficients.
The published method used full and interrupted adiabatic passages, followed by removal of unwanted-state atoms, to make selection more uniform. Its measured number ratios were near 0.506, not an instruction to replace every measured ratio by exactly one half. The preparation comparison and its residual limitations belong in the correction.
Read the evidence: F. Pereira Dos Santos et al. (2002), Controlling the cold collision shift in high precision atomic interferometry — arXiv:physics/0209010v1
Extrapolation also amplifies noise
At an exact one-half density ratio, the intercept is twice the low-density frequency minus the high-density frequency. If those two estimates have independent equal standard uncertainty, the intercept’s standard uncertainty is larger by the square root of five. Shared oscillator noise instead requires their covariance.
The denominator also makes ratios near one poorly conditioned: nearly equal densities supply little extrapolation leverage. More than two densities can test curvature and changes of preparation. Two points alone always define a line, so they cannot establish that a linear density law is adequate.
Read the evidence: F. Pereira Dos Santos et al. (2002), Controlling the cold collision shift in high precision atomic interferometry — arXiv:physics/0209010v1
A correction is a measured procedure
The Ramsey Lab illustrates finite-count interrogation and the Rabi Lab illustrates population transfer. Neither models a colliding cesium fountain or performs the historical adiabatic-density correction. Their records should not be combined and labeled a new experimental uncertainty budget.
For the real inference, retain the frequency difference, number-ratio calibration, preparation settings and tests of the common-coefficient assumption. The result is a zero-density estimate under those conditions, while the atoms used to obtain it were measured at finite density. That distinction is what makes the correction reviewable.
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