---
title: "Cold-atom thermometry: read the flight, then infer the temperature"
description: "What must be known before a spreading cloud or a delayed fluorescence peak becomes a thermometer?"
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language: "en"
page_type: "story"
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last_modified: "2026-10-18"
---

[Quantum physics: reading the evidence](https://sandboxphysics.com/stories/quantum-evidence/)

Nobel A · Evidence and control

# Cold-atom thermometry: read the flight, then infer the temperature

Sandbox Physics · 2026-10-18

What must be known before a spreading cloud or a delayed fluorescence peak becomes a thermometer?

## A detector sees arrivals

Lett and colleagues released sodium atoms from optical molasses in 1988 and compared four thermometers: time of flight, an upward fountain, a sideways shower and release–recapture. Their agreement supported temperatures below the simple Doppler prediction. The methods sampled different aspects of the released cloud.

In the time-of-flight method, fluorescence was recorded as atoms crossed a probe below the source. Arrival time therefore depended on source–probe separation, initial positions, velocities and gravity. A late peak was evidence to interpret through that geometry, rather than a direct display of temperature.

Read the evidence: P. D. Lett et al. (1988), Observation of Atoms Laser Cooled below the Doppler Limit, PRL 61, 169–172

## Subtract the motion of the center

For an illustrative horizontal measurement, let position and velocity refer to one axis at release. Their standard deviations describe spread around the moving center. Under force-free ballistic motion, taking the variance of the initial position plus velocity times elapsed time gives the identity below.

The covariance term measures initial position–velocity correlation. Uniform gravity changes a vertical centroid but leaves this central variance identity unchanged; a force gradient does not. Only for an appropriate thermal velocity distribution can the velocity variance be converted into a directional kinetic temperature using the atomic mass and Boltzmann constant.

$$
\sigma_x^2(t)=\sigma_{x,0}^2+2t\,\operatorname{Cov}(x_0,v_0)+t^2\sigma_{v,0}^2
$$

Read the evidence: P. D. Lett et al. (1988), Observation of Atoms Laser Cooled below the Doppler Limit, PRL 61, 169–172

## Two invented images make a checkable example

Consider an invented uncorrelated cloud with an initial root-mean-square width of 0.50 millimetres and a velocity standard deviation of 0.10 metres per second. After 10 milliseconds its width is about 1.12 millimetres; after 20 milliseconds it is about 2.06 millimetres. These are standard deviations, not full widths at half maximum.

With a sodium mass approximated by 23 atomic mass units, this chosen velocity spread corresponds to about 27.7 microkelvin. Recovering the velocity variance from the two squared widths gives the same value. Dividing either width by its flight time instead overestimates it because the initial cloud was already extended.

Read the evidence: P. D. Lett et al. (1988), Observation of Atoms Laser Cooled below the Doppler Limit, PRL 61, 169–172

## A contracting cloud need not be cooling

Imagine atoms initially moving toward the cloud center. Their position–velocity covariance is negative, so the cloud first contracts and later expands even when every atom keeps a constant horizontal velocity. A single image near the focus would look exceptionally compact without showing any loss of kinetic energy.

Multiple flight times can separate constant, linear and quadratic contributions to the squared width. This is an inference choice: setting the covariance to zero reduces the fit parameters but adds a preparation assumption. A small residual does not test that assumption unless the sampled times and uncertainties could have revealed the missing term.

Read the evidence: P. D. Lett et al. (1988), Observation of Atoms Laser Cooled below the Doppler Limit, PRL 61, 169–172

## The probe selects a record

A finite probe or camera field can miss the fastest atoms. The measured width then describes the accepted subset. Background subtraction, spatial magnification and the optical response must also be specified before converting a pixel width into a velocity distribution.

The original experiment compared complementary geometries and included the initial distribution and gravity in its analysis. That gives a concrete reading task: identify the recorded signal, the integration over unseen coordinates, and the parameters fixed by separate measurements. Different thermometers can share calibration assumptions even when their signals differ.

Read the evidence: P. D. Lett et al. (1988), Observation of Atoms Laser Cooled below the Doppler Limit, PRL 61, 169–172

## Return to the cooling question

The existing cooling Lab helps compare damping, diffusion and capture in its declared force model. It does not acquire the four historical thermometry records, and its temperature readout should not be treated as an independent experimental calibration.

The practical question to carry forward is whether an observed change belongs to the velocity spread, the cloud center, the accepted population or the thermometer model. The companion Sisyphus reading then asks which internal-state mechanism can account for cooling that a two-level Doppler model misses.

Read the evidence: P. D. Lett et al. (1988), Observation of Atoms Laser Cooled below the Doppler Limit, PRL 61, 169–172 J. Dalibard and C. Cohen-Tannoudji, Laser cooling and trapping of neutral atoms — author-hosted review, Sections 3–5

## Continue with the models

These are existing teaching Labs. Their simulated records are separate from the historical observations discussed above.

- [Compare cooling forces and velocity spread](https://sandboxphysics.com/labs/laser-cooling-magneto-optical-trap/)

## Sources and reading notes

We read the 1988 Lett et al. PRL report, especially its four ballistic thermometers and Figures 1–4, through the APS PDF text. Direct local PDF retrieval returned 403. The covariance identity and numerical cloud below are our own illustrative calculation, not a fit to the sodium records.

1. [P. D. Lett et al. (1988), Observation of Atoms Laser Cooled below the Doppler Limit, PRL 61, 169–172](https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.61.169)
2. [J. Dalibard and C. Cohen-Tannoudji, Laser cooling and trapping of neutral atoms — author-hosted review, Sections 3–5](https://pro.college-de-france.fr/jean.dalibard/publi2/laser_cooling.pdf)

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