Start with the four thermometers in the 1988 paper
Open Lett and colleagues’ Observation of Atoms Laser Cooled below the Doppler Limit, published in Physical Review Letters in July 1988. Its central task is to turn the motion of released sodium atoms into a temperature. Read the four ballistic methods together: time of flight, an upward fountain, a sideways shower, and release followed by recapture. Each takes the atoms out of the cooling light and asks where they go.
The paper reports temperatures as low as 43 microkelvin with an uncertainty of 20 microkelvin, well below the simple Doppler prediction of 240. Following the measurement geometry explains why the group could trust this surprising result. Keep a sketch of the source cloud, probe region and direction of gravity beside each signal.
Read the evidence: P. D. Lett et al. (1988), Observation of Atoms Laser Cooled below the Doppler Limit, PRL 61, 169–172
Read an arrival trace through its geometry
In the time-of-flight arrangement, a probe below the source produces fluorescence when falling atoms cross it. Time runs along the trace, and the signal counts atoms reaching the lit region. The peak is shaped by the source-to-probe distance, the initial cloud size, the initial velocities and gravitational acceleration.
Start by asking when an atom released with zero vertical velocity would arrive. Faster and slower initial velocities change that time, but so does starting higher or lower. A broad arrival trace becomes a velocity distribution only after these spatial and timing effects have been included. The fountain and shower arrangements change which parts of that motion are easiest to observe.
Read the evidence: P. D. Lett et al. (1988), Observation of Atoms Laser Cooled below the Doppler Limit, PRL 61, 169–172
For cloud images, follow the width around the center
A related imaging calculation is simplest along a horizontal axis with no force after release. Subtract the moving center from each image before measuring its width. Uniform drift moves that center, while the spread of velocities makes the cloud broaden. Let the width mean its standard deviation, in metres, and let time be measured in seconds.
The identity below separates the initial squared width, a term linear in time and a term quadratic in time. The subscript zero denotes release. The covariance measures whether atoms farther to one side tend to move toward or away from that side; its units are square metres per second. The velocity variance has units of square metres per square second. For a thermal distribution, multiplying it by the atomic mass and dividing by Boltzmann’s constant gives the temperature along this axis.
Read the evidence: P. D. Lett et al. (1988), Observation of Atoms Laser Cooled below the Doppler Limit, PRL 61, 169–172
Keep the initial size in a worked example
Take an invented, uncorrelated cloud whose initial width is 0.50 millimetres and whose velocity standard deviation is 0.10 metres per second. After 10 milliseconds its width is about 1.12 millimetres; after 20 milliseconds it is about 2.06. These are standard deviations rather than full widths at half maximum.
For sodium with mass about 23 atomic mass units, that velocity spread corresponds to about 27.7 microkelvin. Subtracting the two squared widths and dividing by the difference of squared flight times recovers the chosen velocity variance. Dividing either width by its own flight time overestimates it: part of the observed width was present before any flight occurred.
Read the evidence: P. D. Lett et al. (1988), Observation of Atoms Laser Cooled below the Doppler Limit, PRL 61, 169–172
Use several times to expose a focus
Suppose the atoms start out moving inward. The cloud first shrinks, passes through a focus and expands again, even though every atom keeps the same horizontal velocity. Initial position and velocity are then negatively correlated. A small photograph near the focus can coexist with substantial random kinetic energy.
Several flight times allow the constant, linear and quadratic terms in the squared width to be fitted separately. If the linear term is fixed to zero, the preparation must justify that choice. Uniform gravity accelerates every atom equally and shifts the vertical center. Spatially varying forces can also change the width and require another motion model.
Read the evidence: P. D. Lett et al. (1988), Observation of Atoms Laser Cooled below the Doppler Limit, PRL 61, 169–172
Finish by checking which atoms reached the detector
A small probe region or camera can miss the fastest atoms. Their missing tails make the retained distribution look narrower. Check the field of view, background subtraction, magnification and optical blur before interpreting a width. Several thermometers help most when their different geometries test effects that one arrangement could conceal.
For the next measurement, write down the recorded signal, the release size, the forces during flight, the detection region and the separate calibration measurements. Then identify which part of the fit measures random motion and which part measures the center. For a nonthermal cloud, a one-axis kinetic variance remains a useful result with its own definition; a single equilibrium temperature would need further justification.
Read the evidence: P. D. Lett et al. (1988), Observation of Atoms Laser Cooled below the Doppler Limit, PRL 61, 169–172J. Dalibard and C. Cohen-Tannoudji, Laser cooling and trapping of neutral atoms — author-hosted review, Sections 3–5