Time-of-flight and correlation imaging

After forces and collisions become negligible, each particle moves from its initial position with its initial velocity. At long flight times the displacement from velocity can exceed the initial cloud extent. Only in that far-field limit does the expanded density directly reproduce the initial momentum distribution, with a Jacobian needed to conserve particle number. A Fourier distribution defined in wavenumber has a different Jacobian from one defined in momentum; the normalization must be stated.

For a noninteracting classical thermal sample with no initial position–velocity covariance, the spatial variance is its initial variance plus flight time squared times velocity variance. This permits temperature extraction without assuming that the initial cloud was a point. Collective flow introduces covariance or an additional velocity component. A dense condensate can convert interaction energy into anisotropic expansion, so its final width is not automatically a thermal velocity width. Changing scattering length before release can help but also changes the preparation protocol.

In weak-probe absorption imaging, the logarithm of reference intensity divided by transmitted intensity is optical depth. It equals an absorption cross section times density integrated along the imaging axis under ideal conditions. Probe detuning, polarization, transition strength, saturation, and optical pumping affect that cross section. Stray light and camera offsets can corrupt the logarithm, particularly in optically thick regions. Independent atom-number calibration is necessary before using a density-sensitive interaction or loss law.

The imaging point-spread function broadens narrow structures and averages short-distance fluctuations. A Gaussian thermal fit is appropriate only in its stated classical regime; quantum-degenerate Bose and Fermi distributions have different shapes. Fitting the thermal wings can reduce condensate contamination, but limited signal and interactions still need assessment. A bimodal distribution or interference peaks provide evidence for coherence or condensation within a model. Superfluidity additionally concerns response and phase rigidity, so one visually narrow peak does not establish every many-body property.

Repeated images allow density–density correlations after removing the smooth mean profile and accounting for self-counting and detector noise. Bosonic bunching and fermionic antibunching reflect exchange statistics, while additional correlations can reveal lattice order or pairs. Column integration and finite resolution reduce observable contrast. Shot-to-shot number variation and common technical noise can generate correlations unrelated to the desired quantum effect. Report normalization, spatial binning, noise subtraction, and an appropriate uncorrelated reference alongside any correlation peak.

Lifetime measurements supply another diagnostic because three-body recombination weights the cube of local density. The atom-loss coefficient must include a declared counting convention, including how many atoms leave per event and any local correlation factors. As density changes, loss is generally nonexponential, even at fixed volume. Trapped profiles, heating, and changing correlations can further modify the curve. A fitted loss coefficient and a fitted temperature are complementary evidence, and neither replaces calibration of the evolving density.