How did an expanding cloud reveal that atoms shared one quantum state?
Laser cool and trap atoms
Remove energetic atoms
Release the cloud and image absorption
Compare thermal and condensed components
The revealing feature of an early condensate image is a change in shape: a compact central component appears inside a much broader cloud. In a dilute gas, that extra component can signal that a large number of atoms have entered one quantum state, a shared quantum pattern of motion. Bose–Einstein condensation means this shared occupation; it does not mean that the atoms have frozen into a solid or stopped moving.
The 1995 experiments had to establish that interpretation from preparation and measurement. The Cornell–Wieman collaboration worked with rubidium, and the MIT group led by Ketterle worked with sodium. Their samples were released from traps and photographed after expansion. The camera detected light affected by atoms, so the famous density pictures already contained several steps of physical inference. To read them, start with what was actually controlled: the trap, cooling sequence, expansion time and probe light.
An atom has a matter-wave length scale as well as a position. Cooling reduces the typical momentum spread and increases the thermal de Broglie wavelength. Quantum statistics becomes important when that length is comparable with the separation between atoms. The relevant comparison is therefore temperature together with density, not a temperature number alone.
The wavelength varies inversely with the square root of atomic mass times absolute temperature. The separation can be estimated from the volume available per atom: a denser gas leaves less room between neighbours. Consider an illustrative rubidium-87 gas at two hundred nanokelvins, two ten-millionths of a degree above absolute zero. Its wavelength is about four tenths of a micrometre. At ten trillion atoms per cubic centimetre, the mean separation is about half a micrometre. Cooling that hypothetical gas to fifty nanokelvins doubles the wavelength if density is held fixed. Temperature and separation together determine when matter waves overlap appreciably.
Sources: [3]
Laser cooling prepares a large, slow cloud, but the final approach to condensation used evaporation. A trap holds low-energy atoms while an adjustable escape threshold removes more energetic ones. Elastic collisions redistribute the remaining energy. Lowering the threshold in stages can then cool the survivors, much as evaporation cools a liquid.
This requires a balance. A tightly confined cloud collides often enough to re-equilibrate, but unwanted collisions and background gas can remove atoms or heat the sample. The MIT sodium experiment also had to suppress spin-flip losses near a zero of the magnetic trapping field. A focused repulsive laser beam blocked access to that region. Its published report describes a millionfold rise in phase-space density, the concentration of atoms in combined position and momentum space, during seven seconds of evaporation and condensates containing up to roughly half a million atoms. The required temperature changes with density, atomic species and trap shape; these numbers record how the sodium apparatus crossed its threshold.
Sources: [2]
A short probe pulse passes through the released atoms. Resonant scattering removes photons from the light reaching the camera. A reference image without atoms and a dark image allow the transmitted intensity to be normalized. Taking the negative logarithm of the transmission ratio gives optical depth. With a known effective scattering cross-section, optical depth measures column density: the number of atoms integrated along the viewing direction per unit area.
In the formula, the intensity with atoms is divided by the reference intensity without atoms; both are corrected for the dark background. Optical depth is dimensionless. The effective cross-section is an area measuring how efficiently one atom removes probe photons. For a suitably weak probe and a calibrated transition, multiplying that area by column density gives optical depth. Saturation, stray light and finite resolution can distort a dense central region. Releasing the gas reduces the optical density and enlarges features that were difficult to resolve inside the trap.
Sources: [3]
A thermal gas has atoms spread over many motional states. After release, their different velocities produce a broad distribution. Crossing the condensation threshold adds a component with a different width and shape. In the sodium experiment, the thermal part was approximately isotropic after expansion and the condensate formed an elliptical core. Tracking that change as evaporation proceeded made the interpretation more specific than identifying a single anomalously narrow photograph.
One can first use the outer wings, where the condensate contributes little, to constrain the thermal population. The central excess then estimates the condensed fraction using an appropriate interacting-gas profile. This separation carries uncertainty: the thermal density beneath the centre is partly inferred, and imaging errors affect the two fitted components differently. Repeating measurements at several temperatures and expansion times tests whether one description of the two populations accounts for the whole sequence.
For freely moving, noninteracting atoms, long time of flight maps initial velocity onto final position. An initially small cloud then acts like a slow-motion velocity analyser. A dense condensate has an additional energy reservoir: repulsive interactions. When confinement disappears, that interaction energy can become directed expansion energy.
The direction that was most tightly confined can expand fastest. An elongated trapped condensate can therefore change its aspect ratio and eventually appear broader along its initially narrow direction. Castin and Dum developed scaling solutions for an interacting condensate in a changing harmonic trap, with a mean-field approximation that describes each atom through the average interaction with the surrounding gas. Their analysis explains why interpreting every released image as the untouched initial momentum distribution would fail. The experimenter must compare the image with the appropriate expansion dynamics, including the trap’s anisotropy.
Sources: [4]
The cloud can also maintain phase coherence: a stable relationship between where its matter waves are in their cycles at different positions. In 1997 the MIT group released two separated condensates and allowed their matter waves to overlap. The absorption pictures contained fringes. Their spacing changed with the separation and the expansion time in the way expected for interference of expanding matter waves.
In the simple long-time, free-expansion estimate, the fringe spacing is Planck’s constant times flight time divided by atomic mass and initial separation. Larger separation makes finer fringes; a longer flight enlarges them. This is a testable spatial prediction, not just a claim that the image looks wavelike. Two independent condensates need not begin with a reproducible relative phase, so the position of the fringes can vary between repetitions. Separate Bragg-spectroscopy measurements later probed momentum width and coherence with a controlled optical momentum transfer. Density, interference and spectroscopy each supplied a different constraint.
Dilute atomic condensates made a quantum fluid whose density, confinement and preparation could be varied deliberately. The dilute limit also gave theory a useful starting point: atom spacing is large compared with the interaction range, even though quantum statistics dominates the collective state. This combination made the gas a laboratory for confronting many-body calculations with measured profiles and dynamics.
The original evidence joined three observations: a new narrow component appeared, its population changed with cooling, and its expansion followed the dynamics of an interacting gas. Interference later revealed the cloud’s wave phase, and momentum-sensitive spectroscopy resolved another aspect of its motion. Together these experiments made an initially surprising photograph into a controllable quantum fluid, with a preparation that could be changed and a response that could be measured again.
Original collaboration’s NIST publication record for the rubidium condensate below two hundred nanokelvins.
The optically plugged trap, seven-second evaporative cooling, bimodal distribution and anisotropic expanding core.
The experimentalists’ detailed account; sections on absorption imaging, phase-space density, two-component fits and their limits were consulted.
Scaling dynamics in a changing harmonic trap; distinguishes interacting expansion from a direct ballistic momentum map.
Author-hosted original paper on expanding-cloud interference, fringe spacing and relative phase.
Original momentum-sensitive spectroscopy experiment; separates momentum width, mean-field energy and coherence information.
You are hereHow expanding atom clouds revealed Bose–Einstein condensation
Condensed matter · Reading path
Each experiment from this article that another series also uses, followed by the articles there.