= Solution
\b[<Evaporative cooling of atoms> selectively removes atoms with unusually high total <energy>, then <elastic collisions> rethermalize those remaining.] Loss of particles alone is insufficient: the lost atoms must carry more than the mean <energy> per particle if the <temperature> is to fall. The process is open and dissipative, so it can increase the <phase-space density of an atomic gas> in the remaining sample while reducing its atom number. As the <temperature> falls, the cutoff must be lowered to keep replenishing a useful escaping tail; a fixed high cutoff eventually gives negligible evaporation.
An experimental sequence first loads laser-cooled atoms into a conservative trap and suppresses resonant-light heating. In a trap using <magnetic trapping of neutral atoms>, a <radiofrequency evaporation knife> drives a <Zeeman splitting> transition from a trapped weak-field-seeking state to an untrapped state on a surface where the splitting matches the radiofrequency <photon> <energy>. Only atoms energetic enough to reach that surface escape. Lowering the radiofrequency cutoff progressively removes less energetic atoms. Alternatively, lowering an <optical dipole trap>'s depth permits energetic atoms to spill out. This changes confinement as well as the escape cutoff, so it can reduce the collision rate. Radiofrequency-selective evaporation and the need for favourable elastic-collision and background-loss rates are documented in the original condensate experiments.<Nobel Lecture: Bose-Einstein condensation in a dilute gas, the first 70 years and some recent experiments by Eric A. Cornell and Carl Wieman>[pp. 880–882, elastic/background collisions, magnetic confinement, and radiofrequency evaporation]{ref}
The <elastic collision> rate $\Gamma_{\rm el}\sim n\sigma_{\rm el}\bar v$ must be large enough for <elastic rethermalization of a trapped atomic gas> throughout evaporation, and much faster than unwanted loss and heating. A clean vacuum reduces background collisions; suitable trapping states suppress inelastic spin-changing loss, and <density> must not make <three-body recombination> dominant. The cutoff ramp should allow equilibration without taking so long that background losses dominate. A large <evaporation truncation parameter> $\eta=\epsilon_{\rm cut}/(k_BT)$ selects energetically valuable atoms, but an excessively large value suppresses their escape rate. A very small value discards too many useful low-energy atoms. If <elastic collisions> become more frequent as evaporation proceeds, the resulting <runaway evaporative cooling> can improve efficiency; it is not automatic in a weakening optical trap.
The <energy selectivity for evaporation to increase phase-space density> can be made precise in a classical three-dimensional harmonic trap with \i[fixed <frequencies>]. <Equipartition> gives $E=3Nk_BT$. If each escaping atom removes mean <energy> $\epsilon_{\rm ev}$, <energy> balance is $dE=\epsilon_{\rm ev}\,dN$ with $dN<0$, so
$$
\alpha_{\rm ev}\equiv\frac{d\ln T}{d\ln N}=\frac{\epsilon_{\rm ev}/(k_BT)-3}{3}.
$$
The peak <harmonic-trap phase-space density> is $\mathcal D=N(\hbar\bar\omega/(k_BT))^3$. Therefore
$$
\boxed{\frac{d\ln\mathcal D}{d\ln N}=4-\frac{\epsilon_{\rm ev}}{k_BT}.}
$$
Cooling requires $\epsilon_{\rm ev}>3k_BT$; increasing $\mathcal D$ despite atom loss requires the stronger condition $\epsilon_{\rm ev}>4k_BT$. These inequalities assume rapid thermalization and no work from changing trap <frequencies>. They distinguish lower <temperature> from genuinely improved <quantum degeneracy>. Eventually <quantum statistics> and inelastic processes require a refinement of this classical estimate. \b[Efficient evaporation combines energy-selective loss, rapid elastic redistribution, and low unwanted loss; it can bring a precooled gas to <Bose-Einstein condensation>.]
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