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In a dilute gas, Bose-Einstein condensation makes a single one-particle state hold a substantial fraction of the bosons. This is a population effect: cooling reduces the number that the excited energy eigenstates can accommodate at thermal equilibrium. The surplus accumulates in a Bose-Einstein condensate. For an ideal gas the occupied mode is the trap ground state; interactions can deform that mode, as described by the Gross–Pitaevskii equation. Dilute does not mean classical: a small number density can still give overlapping thermal de Broglie wavelengths at sufficiently low temperature.

The appeal is that changing the trap changes a calculable Quantum Hamiltonian, while absorption imaging makes its population distribution visible. The Bose condensation in a three-dimensional harmonic trap below turns density of states into a prediction for a critical temperature and condensate fraction. These are separate questions from measuring matter-wave coherence or superfluidity.

Eric A. Cornell and Carl E. Wieman's joint lecture emphasizes the competition between useful elastic collisions, which restore thermal equilibrium during evaporative cooling, and collisions that remove or heat the trapped sample. A long trap lifetime alone is insufficient: reequilibration must keep up with evaporation.^[ref]

The rubidium experiment used laser cooling, a magneto-optical trap and magnetic trapping of neutral atoms with a rotating bias field. Radio frequency transitions removed energetic atoms; elastic collisions restored thermal equilibrium. Before absorption imaging, the trap was decompressed, then nearly released with a gradient supporting against gravity. Varying the final evaporation cut revealed a narrow anisotropic core within an isotropic thermal cloud. The broad component supplied a simultaneous comparison through the same optics. Resolution and size calibration limited inference; interactions broadened the core relative to an ideal gas.^[ref]

The 1995 sodium experiment combined laser cooling with a magnetic quadrupole trap and a repulsive optical plug protecting its field zero. Radio frequency evaporation preceded release and absorption imaging after time of flight. Lowering the evaporation cut produced broad thermal wings and a narrow core. Moving the plug changed the core's expansion orientation while the thermal distribution remained spherical: a control connecting the feature to confinement. Overlap of the two trapped pockets and residual switch-off acceleration limited quantitative shape comparisons; repulsive interactions affected expansion.^[ref]

In 2001, Andrew G. Truscott, Randall G. Hulet, and collaborators observed degeneracy pressure in trapped lithium. A Zeeman slower fed a dual magneto-optical trap; optical pumping prepared atoms for a cloverleaf apparatus for magnetic trapping of neutral atoms. Microwave evaporation removed energetic lithium-7 bosons; elastic collisions cooled lithium-6 fermions by sympathetic cooling.^[ref]

Sequential absorption imaging recorded both trapped clouds on a camera. Fits to axial profiles used the boson cloud as a thermometer. At common temperature, the nearly equal magnetic trapping potentials provided a comparison: the clouds had similar sizes near $T/T_F=1$, but the fermion cloud remained broader as cooling reached $T/T_F=0.25$. Figure 3 compared its radius with ideal Fermi gas and classical predictions.^[ref]

Varying the wait before imaging tested thermal equilibrium. Finite imaging resolution restricted the analysis to the axial direction; fits neglected interaction-induced density shifts. Cooling eventually suffered from the dwindling boson heat capacity. The result supports quantum occupation effects, without directly resolving individual occupied states or reaching zero temperature.^[ref]

These were neutral alkali atoms in a spatially varying trap. The Fermi energy of a model electron gas instead uses uniform Electrons, two equally populated spin states, and the Electron mass. The measured trapped-atom radii are not measurements of that electron model's $p_F$ or $E_F$; comparing their numerical Fermi energies requires the appropriate particle mass, internal-state populations, and confining potential.

A broad component plus a narrow central component can have one local maximum while still displaying two distinct width scales. This is what “bimodal” means in this imaging context; it does not require two separated peaks. To isolate the population effect, construct an original normalized line profileBoth terms are normal distributions; their integrals are one, so the component areas are $1-f_0$ and $f_0$. With $f_0=0.488$, the narrow component has central height $f_0/(\sqrt{2\pi }0.20)\simeq 0.9734$, while the broad component contributes $(1-f_0)/\sqrt{2\pi }\simeq 0.2043$. A tall central feature therefore need not contain most of the particles. Holding these widths fixed deliberately isolates changing weights; it does not model how temperature, interactions or camera resolution determine actual widths.

For absorption imaging in the weak-probe regime with negligible saturation, the optical depth is $\mathrm{OD}(x,z)=-\ln [I(x,z)/I_0(x,z)]={\sigma }_{\mathrm{abs}}{n}_{\mathrm{col}}(x,z)$, where $I_0$ is the reference light intensity without atoms, $I$ the transmitted light intensity, ${\sigma }_{\mathrm{abs}}$ an effective absorption cross section, and $n_{\mathrm{col}}$ the column density. Integrating $n_{\mathrm{col}}$ over both image coordinates gives the particle number; a central pixel height alone does not. Finite resolution can obscure a narrow component, and saturation invalidates this simple logarithmic calibration.

In 2006, Zoran Hadzibabic, Peter Krüger, Marc Cheneau, Baptiste Battelier and Jean Dalibard split a cooled rubidium-87 cloud into two independent planar gases using an optical lattice. After release, the expanding sheets overlapped; absorption imaging recorded their matter-wave interference. Smooth fringe waviness revealed thermal phase fluctuations, while sharp dislocations indicated isolated vortices. The initial evaporative-cooling endpoint controlled the preparation; central fringe contrast served as a degeneracy proxy because direct thermometry in the lattice was difficult.^[ref]

Loss of extended coherence coincided with increased vortex signatures. The optical trap and finite cloud produced a crossover; tightly bound pairs were unresolved, and short-wavelength modes could contaminate high-temperature counts. These were observations of the vortex-unbinding mechanism, not a direct measurement of phase stiffness or a sharp homogeneous universal stiffness jump. The authors left the sample's superfluidity open.^[ref]