Evaporative cooling and phase-space density

Define the thermal de Broglie wavelength using Planck's constant divided by the square root of two pi times mass and thermal energy. Its cube multiplied by density is a dimensionless phase-space measure. The numerical convention matters when comparing a quoted threshold. In a uniform ideal three-dimensional Bose gas, condensation begins at a phase-space density of about 2.612, rather than exactly one. The rough statement that wavelengths overlap describes a scale, not the exact threshold.

A harmonically trapped gas has an inhomogeneous density and a different global critical-temperature relation. In the classical thermal regime, its central phase-space density can be written using atom number and the geometric mean angular trap frequency. This formula is useful before degeneracy, but should not be extrapolated through condensation as a complete description of the density profile. Fermionic degeneracy instead compares temperature with the Fermi temperature for a declared component density and spin multiplicity.

Atoms escaping over a finite barrier or through an energy-selective radio-frequency knife remove energy above a chosen cutoff. Elastic collisions repopulate the high-energy tail and produce a new near-thermal distribution. The cutoff relative to thermal energy controls both how many atoms leave and how much energy each removes. Very aggressive truncation can leave a nonequilibrium cloud, while an extremely high cutoff removes too few atoms before background loss becomes important.

Efficient evaporation needs a favorable ratio of elastic rethermalization to inelastic or background loss. At low energy, identical bosons in one internal state have an s-wave cross section twice the distinguishable-particle result. Identical spin-polarized fermions cannot use s-wave collisions, so a second spin component or another species is often needed for efficient cooling. Close to a scattering resonance, finite collision energy limits the cross section, and simply extrapolating its zero-energy square-law growth is invalid.

At fixed harmonic frequencies, central phase-space density scales as number divided by temperature cubed. A modest temperature reduction can therefore compensate substantial number loss. If lowering optical power also weakens confinement, the trap frequencies fall and reduce this gain. Adiabatic decompression of an ideal classical harmonic cloud lowers its temperature in proportion to trap frequency and preserves phase-space density at fixed number. It is cooling in a thermometer sense without the same entropy reduction per particle as selective evaporation.

Three-body recombination and light-assisted losses often remove atoms from the densest part of the cloud and can heat the survivors. A high collision rate is therefore not sufficient by itself: increasing density may improve rethermalization while worsening inelastic loss. Report evaporation efficiency over a stated trajectory of number, temperature, and confinement, together with the thermometry and density calibration. Reaching a fitted temperature below a target does not alone establish condensation or superfluidity.