= Solution
<evaporative cooling (atomic physics)> removes atoms selectively from the high-energy tail of a trapped gas. The remaining sample has less <energy> per atom; <elastic collisions> then redistribute <energy> and restore a thermal distribution at lower <temperature>. Random atom loss would generally remove average <energy> and would not provide the same cooling mechanism. Repeated <energy> selection and rethermalization, rather than one removal event, is the essential cycle.
Using the <evaporation truncation parameter> $\eta$, let the escape threshold be $\epsilon_{\rm cut}=\eta k_BT$. As the gas cools, a fixed threshold is populated by fewer atoms, so evaporation stalls. In <forced evaporative cooling> the threshold is lowered during the sequence. In a magnetic trap, a <radiofrequency evaporation knife> flips trapped weak-field-seeking atoms into untrapped <spin> states where the Zeeman splitting matches the <photon> <energy>. Energetic atoms reach that resonance surface; sweeping the radiofrequency lowers the <energy> knife. In an optical dipole trap, lowering the trap depth offers a different implementation, although it usually also weakens confinement.
A simple <evaporative cooling energy balance> shows when atom loss is useful. For a classical gas in a three-dimensional harmonic trap of fixed frequencies, equipartition gives $E=3Nk_BT$, counting both kinetic and <potential energy>. Suppose escaping atoms remove average <energy> $\epsilon_{\rm ev}=(\eta+\kappa)k_BT$, with $\kappa$ a distribution-dependent excess above the threshold. For $dN<0$,
$$
d(3Nk_BT)=\epsilon_{\rm ev}\,dN,\qquad
\alpha\equiv\frac{d\ln T}{d\ln N}=\frac{\eta+\kappa-3}{3}.
$$
Thus preferential removal cools when the removed <energy> exceeds $3k_BT$. More stringently, cooling must raise the <phase-space density of an atomic gas>, not merely lower the <temperature>:
$$
\mathcal D=n_{\rm peak}\lambda_T^3
=N\left(\frac{\hbar\bar\omega}{k_BT}\right)^3,\qquad
\lambda_T=\frac{h}{\sqrt{2\pi mk_BT}}.
$$
For this <harmonic-trap phase-space density>, with fixed geometric-mean trap <frequency> $\bar\omega$, $d\ln\mathcal D/d\ln N=1-3\alpha$. Therefore \b[phase-space density rises during atom loss if $\alpha>1/3$]. The numerical factors here belong to the stated harmonic-trap model, not every evaporative apparatus.
\Image[/past-exam-of-the-atomic-and-optical-physics-course-of-the-university-of-cambridge/2016/paper-1-evaporation.png]
{title=Energy-selective loss lowers the mean <energy> before rethermalization}
{description=Original classical harmonic-trap <energy> distribution, with an explicitly illustrative dimensionless cutoff. The post-cut distribution is truncated and is not yet a <thermal equilibrium> distribution.}
There must be enough <elastic collisions> to repopulate the tail: $\Gamma_{\rm el}\sim n\sigma\bar v$ should exceed damaging loss and heating rates, and the ramp should allow rethermalization. Increasing density can improve <elastic collisions> but also enhances inelastic and three-body losses. Very rapid removal can leave a nonequilibrium cloud; very slow removal exposes it to background-gas losses. Lowering an optical trap can reduce density and collision rate, so a falling <temperature> is not by itself evidence of efficient evaporation.
The dilute-gas condensation experiments made this tradeoff concrete. <Eric A. Cornell> and <Carl Wieman>'s account explains how selective escape and contraction of a trapped cloud raise phase-space density, while the need for rapid two-body equilibration and slow three-body loss restricts usable density. It also describes expansion imaging as an experimental diagnostic.<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. 876–878, equilibration, selective escape and imaging of expanded clouds]{ref} Useful controls compare different knife ramps and hold times, measure both atom number and expansion <temperature>, and check trap-frequency calibration. Pure adiabatic expansion can cool without increasing phase-space density. \b[Successful evaporation pays an atom-number cost to increase the remaining gas's phase-space occupancy.]
Back to article page
